Any time Reese or Finch refers to the other by their first name. However it's also often an indicator that shit just got real.
From the ending of "Judgement" - "Try the Eggs Benedict, Mr. Reese. I've had them many times."
To put this in context, Finch is obsessed with concealing personal information about himself from everyone, especially Reese. At the beginning of the episode he refused to recommend anything on the menu of that diner because if Reese knows that Finch has a favorite meal there, it means that he's eaten there multiple times and implies that he lives close enough to that diner to make it a practical place for him to have breakfast, and would give Reese a point of data to figure out where Finch lives.
In Season 2 episode 4, the development of their relationship is further demonstrated by the fact that Harold doesn't even hesitate to recommend an entree at a restaurant where Reese is investigating their POI.
By the time 2.08 (Til Death) rolls around, Finch is amused when Reese starts fishing for information and even plays along.
Reese: I didn't know you were a baseball fan, Finch. The Mets? Finch: Not particularly. Reese: Orioles? Finch: I suppose. Reese: The A's. Finch: (catches on, starts smiling) Oakland is fine. And the Cubs, and the Red Sox too. Although I'm not certain my affection for any one of these teams would reveal where I grew up, Mr Reese.
In "Get Carter", it was lampshaded early on that we don't know Detective Carter's first name. Over the course of the episode she demonstrates her ability as a parent, as a cop, as an Army interrogator (via flashbacks), as the contact for a CI, and in general as a badass. At the end of the episode we see that she's not quite badass enough, but Reese saves her life and takes the time to tell her, "Whether you like me or not, Joss, you are not alone."
The dialog between Reese and Finch at the end of "Number Crunch":
Finch: It's not over, John. I'm close, just get to the ground floor.
Reese: No, You stay away. Don't even risk it.
Also doubles as a major tearjerker. Harold noticeably accelerates his car when the call is terminated, showing how he's desperate to save John from dying of his gunshot wounds.
Plus the fact that Finch is willing to risk exposure, death or imprisonment (which would inevitably lead to his death) to save Reese, despite Reese telling Finch not to come and save him.
Finch's occasional usage of his wealth to resolve problems totally unrelated to what the Machine is predicting.
For instance, after one week's POI, an unemployed construction worker, is out of danger (but still unemployed and near bankruptcy), Finch casually mentions to Reese that he owns a construction company that needs a new foreman on a big project the company just landed.
"Risk" Adam Saunders, the POI decides to retire from trading, goes into business with his Uncle's small food company and decides to hire the homeless people he met when he and John were hiding from the people trying to kill them.
Near the end of "Risk", Reese takes a moment to thank his homeless friend Joan for looking after him when he was living on the street. She asks who's looking after him now, and he replies "Someone new" as we see a shot of Finch sitting in the library, listening to the conversation over the phone and he appears touched.
In "Identity Crisis", the normally reserved Finch gets doped up on ecstasy and openly offers to answer any question Reese asks him; Reese refrains from taking advantage of the situation, despite his obvious curiosity about Harold's background.
This troper found the release of Kyle Morrison at the end of the episode very heartwarming, as an innocent man being able to return home to his little girl...*sniff*
"Flesh and Blood" Carter's reunion with her kidnapped son. After holding off Elias for them, her trust in John and Harold is rewarded to the greatest extent possible as they moved heaven and hell to get him back.
In "Many Happy Returns", Finch gives Reese a key as a birthday present but there's nothing to indicate what it goes to. At the end of the episode, Reese learns that it's the key to a brand-new apartment over-looking the park where he plays chess during his free time.
Which incidentally implies that Finch's statement earlier in the episode that he doesn't keep track of what Reese does in his free time was a blatant lie.
Harold lying to John actually becomes heartwarming in hindsight once you understand that he knew what had happened in New Rochelle and was actually trying to spare John the pain of dragging up some highly unpleasant memories.
Not to mention that you could also make an argument that he may have been trying to protect John from himself; while Harold may not know exactly what occurred when John confronted Peter, he likely has a pretty good idea anyways and it could be that he was trying to protect John from having to face that side of himself again.
Finch sees to it that the magazines he owns give his fiancee Grace steady work as a freelance illustrator.
After Reese tells Fusco that he can keep all the money he's making by working undercover in HR, the first thing Fusco thinks to spend it on is his son to get new hockey gear.
After Reese has rescued Finch from Root and returned him to the library, Root calls the person who just ruined her plans to get to the Machine (for now, anyway) ... and thanks him for finding out what happened to her best friend, who had been kidnapped and murdered twenty years previously. It is currently the only humanizing moment for an otherwise amoral character.
In "Masquerade", Finch slowly growing to like Reese's new dog is both Heartwarming and amusing.
Finch: *throws a ball for the dog to fetch while talking to Reese*
In fact, three episodes later, it's quite clear that Bear is Harold's dog, not John's.
Reese's gentle understanding of Finch's PTSD after his abduction by Root, and his gentle but firm insistence that Finch get out of the library.
The birthday scavenger hunt Harold arranges for Grace in "'Til Death".
Harold and Grace in general, really.
Nathan's reaction when he realises Harold is dating someone. With his love life falling to pieces around him, he's genuinely happy that his extremely secretive friend has found someone to love.
Harold's version of marriage therapy for the POI's, utterly understanding and gentle, telling them they're "not monsters" even when they tried to have each other murdered and allowing them to understand how they both contributed to their marriage falling apart and to realise they have a chance to make up for it. And he's doing this while preparing a candlelit dinner for them
In "Shadow Box", Finch comments that Reese seems to be in a good mood; Reese confirms that he is, and says that he actually felt happy when he woke up that morning. "It must be this job." Finch appears touched and responds with a sincere, "Well, I'm glad."
This also doubles as a tearjerker when you realize that this is very likely the first time Reese has been genuinely happy since he and Jessica spent a weekend together in Mexico eleven years ago.
Bear's delight when Reese comes home at the end of "Dead Reckoning" after being away for three episodes.
Not to mention the genuine smile on Reese's face as he accepted Bear's welcome home dog-tackle. Not a smirk, or a lazy smile, but a genuine happy smile. This Troper's heart melted.
Pretty much the entire scene on the rooftop between Finch and Reese in "Dead Reckoning," but especially this bit of conversation:
Reese (a bomb vest strapped to him that will go off in a couple of minutes) trying to get Finch to get out of the danger zone: "This is my past catching up with me; it doesn't concern you."
Finch: "But this moment does. I'm not leaving you here, John, so can we please stop wasting time?"
Fridge Brilliance kicks in when you watch the later episodes Harold's first Only Friend Nathan Ingham was killed by a bomb due to Harold not intervening. Years later with the life of another Only Friend on the line, he's not going to make the same mistake of leaving that person to die and sees this as a chance to make up for letting Nathan's preventable death happen
Also, right before the rooftop scene, Carter and Reese's exchange:
Carter: "John, you don't have to do this."
Reese: "You know I do. Because you'd do the exact same thing."
From "Booked Solid", Reese introduces Mira to Fusco and tells her that she can trust him, and then he looks directly at Fusco afterwards to confirm that he meant it and that he trusts him as well. Considering that he has a tendency to disregard Fusco, that was pretty huge.
Reese: "Mira, this is Lionel; and you can trust him."
At the end of the episode, Mira who's been flustered most of the time and gone through a massive Trauma Conga Line is hired as the new manager of the hotel under Harold's ownership and for once is genuinely smiling.
Finch's speech to Shaw, who is almost insanely paranoid.
Finch: "The world looks like it did ten years ago, but underneath, it's become very strange indeed. An invisible struggle has begun. Most of the people who knew this are already dead. All things being equal, I prefer you didn't join them."
"Proteus": Finch is utterly overjoyed when Carter shoots the POI who was going to kill him and steal his identity. Made all the more sweet when Harold who can't see Carter due to the POI stealing his glasses is asking if it's really her in a rare moment of vulnerability.
At the end of "All In", Finch hires the now out of danger POI to repair the watch he destroyed at the end of "One Percent". It's a two million dollar watch, so repairing it will easily earn Lou the money he needs to buy his late wife's former place of work and ensure that nobody can ever try to turn it into a parking lot again.
"Trojan Horse": Finch and Monica's interactions. She's clearly a woman Finch could get to know better under the right circumstances.
And Reese guessing that Shaw would be hanging around Cole's parents because he would do the same thing in her position, and then going out of his way to track her down and see if she wanted any company. She pretty much tells him to get lost, but as he leaves he tells her that just because they walk in the dark, it doesn't mean they have to walk in it alone.
Extra heartwarming because that was something Kara Stanton told Reese when they first started working together, and he tells Shaw a friend of his shared this with him. Even after everything that's happened, even the bomb vest strapped to his chest, Reese still thinks of Stanton as a friend.
And Finch planting a story in the media which refutes the allegations against Cole and describes him as a hero who died in the line of duty.
Carter being willing to violate her beliefs by hiding a body from Internal Affairs in order to protect Fusco because she believes that he's genuinely reformed. Also Finch lending her Bear for the purpose. Either he knew why she wanted him or trusted her enough to lend him, no questions asked.
The Machine's attachment to Finch is quite cute, in the way only a strange, semi-sentient machine can be; it wants to look after him specifically, despite him trying to teach it not to, sets him up with his future fiance, and generally seems to have an intense amount of interest in him for being "just a machine". His reactions to it definitely range in their cuteness, however.
Reese being willing to forgive Finch selling the laptop to the Chinese and inadvertently setting him and Kara on their mutually antagonistic path.
Corollary to this, Shaw acknowledging that like her, Reese opens up to very few people.
Corollary #2: Shaw trusting Reese, even though she doesn't understand all of his personal history with Finch and the Machine. She's going to make a very formidable addition to Team Machine.
It becomes Harsher in Hindsightwhen you find out that Beecher is innocent and he dies, but Fusco telling Beecher to stay away from Carter was rather sweet. Especially given that the man was about a head and a half shorter and about as intimidating as a growling puppy.
"God Mode": the normally non-empathetic Shaw picking up on Reese's reaction to finding the photo of Jessica in the wall safe.
Also in the same episode, after Root has seen her grand plan explode before her very eyes and is consigned to a mental asylum, she gets a phone call....from the machine in a twisted version of Throw the Dog a Bone.
"Liberty": The Heart To Heart Reese has with the POI, giving him some friendly advice as he's gotten an offer to join the SEALS despite wanting to leave the Navy and settle down.
Throughout the episode, Reese, Harold and Fusco are checking up on the recently demoted Carter to see if she's ok. Reese even complimented her on the uniform.
Reese: "For what it's worth, you're looking kind of badass in that uniform."
"Lady Killer": Shaw, Zoe and Carter bond while going clubbing.
"Razgovor": Gen gives Shaw her grandfather's Order of Lenin to keep on her behalf. While she knows Shaw has APD, she still connects with the older woman on some level, judging from the rather amusingly awkward hug and send-off Shaw gives Gen.
The farewell between Gen and Shaw. For the first time, the former ISA assassin empathizes with someone other than her current Morality Pet and Gen gently informs Shaw she's not a cold hearted monster.
Gen: "You feel things – but the volume is turned way down."
And Finch sponsoring Gen's relocation to a private school and being willing to act as her guardian. It would not be surprising at all if Gen moonlights for Team Machine in another decade.
Also, even after all her recalcitrant refusals to follow Finch's suggestions and orders throughout the episode, Finch still wants Shaw on Team Machine - because Shaw finally understands their cause.
Shaw: Am I fired?
Finch: On the contrary, Miss Shaw. I think you finally got the job.
A quick miss but Finch states to Shaw to go to a doctor that owes them a favor. If this happens to be the man from "Super" that means he was able to become a doctor thanks to the money Finch gave him to save Reese.
Tim and Jason's thirty second phone call at the end of "Mors Praematura".
From earlier in the episode:
Reese: Are you sure you don't need my help on this?
Finch: No, Mr. Reese, tracking Miss Shaw is your top priority.
Reese: Alright. But Finch... I'm here if you need me.
Later in the episode, Reese does indeed appear to save Finch and the PoI Just in Time from being burned alive. Finch's joyous exclamation of "Mr. Reese!" just sells it.
"The Perfect Mark"
After only a few episodes, Laskey goes from antagonist HR mole to someone willing to keep Carter alive even if he has to die to do so
Carter successfully appeals to the dying, morally bankrupt Detective Terney to "be a cop one last time" and successfully acquire the HR leader's true identity.
Carter tells Fusco he was the best partner she ever had.
The reason why Carter has initiated her plan to destroy HR with minimal support from Team Machine. In the event of faliure, she doesn't want John and Harold to be compromised and all the good work they've done over the course of the show to go to waste in a huge contrast to her opinion of them in Season 1.
Even though it nearly leads to her death when she's trying to arrest the head of HR legally, Carter's "World of Cardboard" Speech is this. She wants to give the law a chance to work but knowing it's probably not going to succeed due to Quinn corrupting the system, in the end she decides to do what's right and have Reese drag Quinn out the hard way
Reese and Carter sharing their secrets and fears.
Reese's It Has Been an Honour speech to Carter. He tells her he was nearly tempted to blow his brains out back in Season 1 after he lost Jessica but meeting her for the first time gave him hope.
“A fight with punks on a subway. A cop detained me… and brought me to you. You changed my mind, Joss… You changed me”
Shaw goes and checks up on Fusco and his son during a hockey game and surprises the former in his car. Fusco gives her his sincere thanks for saving his son from being murdered, showing how their antagonistic relationship is slowly changing for the better.
"The Devil's Share"
As John lies wounded in a makeshift hospital bed, Bear sits by his side with his head on his master's lap.
Fusco:That’s just it, I could have been just like you. A bottom feeder who turns on his own kind. For what? Money? Power? I got lucky. I had a partner. She was good for me for a lot of reasons. She reminded me that I could be good again too. I could be a good father, friend, a good cop! I’m not gonna let you undo all the good she did. Carter saved my life. She saved me from myself because she believed in me. I’m not gonna throw all that away on piece of crap like you. Patrick Simmons... you're under arrest.
Also, Fusco is finally free of his dirty cop past by arresting the man, has finally won his redemption, is free from HR and is finally in a position to return to being a non-corrupt NYPD detective which he began to love back in Season 1 but was callously put on hold by John. Things are looking up for Lionel.
Elias and his bodyguard go and visit Officer Simmons. Despite their antagonism, Carter was one of the few people to truly win the man's respect After spelling out how far Simmons made him angry, and noting that while Team Machine is hamstrung by their moral thresholds, Elias isn't, he then has Scarface strangle the man to death. Doubles as a Crowning Moment of Awesome and an epic case of Kick the Son of a Bitch.
Elias: "I don't think she liked me, but I liked her quite a bit. And you killed her. So now it is my responsibility fix the particular problem that is you, Officer Simmons."