This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / The Mentalist
In an extremely rare moment, we see Jane cry in private after a supposed psychic calls him out on the one thing he had been wondering since the murder of his family: Had his daughter been awake when she was killed? She says no. That in itself is a Tearjerker, but it gets better when Van Pelt, who'd argued with him over the existence of an afterlife, closes the door to give him privacy.
Lisbon finding a tape of Jane at the end of "Every Rose has its Thorn." It may double as a Ship Sinking.
Patrick: (on a video recording) I'm looking for someone who... someone I can trust, someone strong. Someone at peace with themselves. Someone better than me. Someone who knows the worst side of me, and still loves me.
Erica: (off camera) Sounds like an amazing woman.
Patrick: She was.
When Jane met an accomplice pretending to be Red John. Especially when he describes to Jane what they smelled like before they died.
The death of the coroner in the episode 'Red Mile'. Watching him slip away while Jane focuses his attention on a simple coin trick is heartbreaking.
Basically any flashbacks of Jane's, including Jane coming home to find his family murdered, or his time in a mental ward.
Jane breaking down, saying that he cannot keep up following Red John and telling Lisbon that he might just stop. This is after burning all the information he has on Red John and basically drink himself to sleep.
"Code Red" has a dying woman's teenage daughter saying good-bye to her over the webcam. It's about as gut-wrenching as you'd expect.
"Devil's Cherry" has Jane meet with his daughter as the result of a belladonna poisoning. No less than three moments may have you in tears: 1. "You are strong, you are loved, you are wise." 2. "Please wait...!" 3. The final scene, where Jane may or may not be drinking Belladonna tea, but there is no doubt what is on his mind. Oh, that music!!
At the end of episode 15, "Scarlett Fever" (itself a bit of a tear-jerker), Lisbon observes that Jane "doesn't like to be thanked" for his work in solving cases and speculates it has something to do with remaining aloof or mysterious: "Who was that masked man, anyway?" But at the end of "Blood Brothers", when Jane sees the victim's family approaching, seems about to retreat, then is spotted by Van Pelt and goes down resignedly and manages to say a few kind words to the family. As she said to her sister: Maybe seeing victim's families reminds him of his own wife and child, who were murdered by Red John. And maybe he hates being thanked because he has to see these people get closure, while the Red John case has never been solved. Cue salty waterfalls.
"Don't you know there are people who care about you, people who need you? You're being selfish and childish and I want you to stop it."
Which is so brilliant because Robin Tunney's acting is perfect and Lisbon sounds on the brink of tears.
"Oh, Lisbon, don't cry." Lisbon - Lisbon, the strongest woman on television - is crying because she thinks she's spinning into madness, and Jane can't do anything to help her. And then he fixes it, because he loves her.
Grace Van Pelt talking a suspect out of suicide, and talking about how her sister killed herself, and how helpless she felt afterwards. Van Pelt isn't usually a big character, just quietly competent and the focus of Rigsby's utter adoration, but here she was so raw. You're reminded just how good all of the leads on this show are.
In "His Red Right Hand", Bosco's death. He finally admits that he loves Lisbon, which manages to be heartbreaking even though he's married, and then she tells him that she loves him too while trying not to cry, and failing, despite being the strongest person on television, and then he dies and she's so heartbroken.
Jane in "Redline", talking to a victim's boyfriend about the death of his girlfriend. The way Lisbon doesn't even stop him... Well, it just shows how subtle this show can be.
"Oh stop. Self-pity'll kill you. Take it from someone who knows."
Jane at his wife and daughter's graves.
A bit related, but notice how in "Rose-Colored Glasses", when Jane is reeling off instruments he thinks Lisbon might've played, he never says piano. His daughter was a piano prodigy.
"Fugue In Red" has a couple, starting with Lisbon's horrified tears as she begs and prays for Jane to live after he's been drowned, and ending with Lisbon "helping" Jane to regain his memory by taking him back to house and showing him the bloody smiley on the wall. Heartbreaking. "I'm happy now. Just let me be happy."
The episode "Ruby Slippers".
"Red Rover, Red Rover" is quite a tearjerker as Jane celebrates the anniversary of his family's deaths by slipping into a breakdown and claiming to give up on seeking his revenge on Red John.
The look on Lisbon's face when she pleads for Jane to let her help him. Her expression alone is bait for waterworks.
So much of Crimson Hat. Jane's fake meltdown, culminating in him getting beat up in an alley. Worse, his fake-killing Lisbon. He hugs her, looks close to tears, and says "Good luck, Theresa. Love you."