While the game has been mostly well-received, a lot of long-term series fans are extremely upset about it's extreme emphasis on action at the expense of the puzzles which made the previous games so popular in the first place.
Captain Obvious Reveal: Many regarding what's happening on the island and explanations for the various phenomena, like the abnormal weather being controlled by Himiko and the Solarii wanting Sam so badly because she can be Himiko's vessel, which were easily guessed by many players. Lara's seemingly belated realizations can be justified by the fact that people playing a video game are more likely to assume the supernatural than someone living those events and the documents that support or even explicitly confirm theories are (mostly) optional, but it still sticks out to a lot of players.
Cliché Storm: There's very little of the story Genre Savvy players won't see coming, especially when it comes to the fate of many of the characters, and you're likely to figure out what exactly is going on on the island before Lara does. Of course Tropes Are Tools, and the story plays out like a check list of The Hero's Journey that works very well for the whole origin story aspects of the game.
Complete Monster: Big BadFather Mathias demonstrates narcissism in its ugliest form. His first diary entry reveals that he saw his fellow survivors from a 1982 plane crash in Yamatai as weak, stupid, and useless liabilities. As a result, he sends them sailing to their deaths simply to prove his theory that escaping the island by boat would be impossible. Eight years later, he forms the Solarii Brotherhood, an Ax-Crazycult that worships Sun Queen Himiko. Mathias, however, sees no value in his followers nor Himiko, as the cult merely serves as a distraction for the other survivors to buy Mathias enough time to find the key to leaving the island by himself. As Lara Croft discovers, Mathias initiates new followers into his cult by having his recruits thrown into a chamber in the Geothermal Caverns and left to fend for themselves for days, even weeks, without water or food. This process destroyed his recruits' humanity and forced many to resort to cannibalism. Mathias also has no problem killing his own followers. When one of his men objects to destroying a bridge to stop Lara from escaping his hideout (which would kill some followers in the process), Mathias shoots the dissenter point blank and asks if there are any other “non-believers.” After noticing that Lara and Conrad Roth's helicopter was shot down by a sudden lightning strike, he attacks the crash site and ultimately tosses a hatchet to kill Roth. Afterwards, Mathias convinces Dr. Whitman to betray the survivors by recapturing Samantha Nishimura, Lara's best friend who is seen by Mathias as his key out of Yamatai. Not long afterwards, Mathias uses Whitman to distract a pair of Oni and leaves his ally to die a brutal death.
Counterpart Comparison: Lara Croft has a lot in common with Oliver Queen from his incarnation in the television show Arrow. Both are young and wealthy and end up accidentally shipwrecked on a mysterious island where it becomes clear that they aren't the only people left there. While originally inexperienced with fighting, both go on to learn numerous skills that come in handy. Lara's primary weapon is even a bow and Oliver Queen is the Green Arrow himself, of course. By the end, they both leave their respective islands changed emotionally and physically and with a purpose (Oliver to fix the corruption in his city, Lara to search for another adventure).
Critical Research Failure: At the start of the game, Lara winds up impaled on a piece of rebar and immediately pulls it out. Do not do this in real life. You'll begin to lose way more blood and possibly do horrible amounts of internal damage.
Escapist Character: This Lara, less than the others. She's relatable on some level, but still larger than life.
Goddamn Bats: Sappers. Their molotov cocktails force you to keep moving from cover to cover, which besides exposing you to enemy fire, makes it very challenging to draw a bead on a target since you're constantly having to reposition to not get blasted. You have to be able to aim and shoot very quickly, (god help you if all you have is the bow and its very low rate of fire) and it certainly doesn't help that they have arms like Major League pitchers and almost perfect aim. Sappers will quickly make gunfights insanely frustrating, especially if there's more than one of them. They get worse when you start running into the ones that carry dynamite, which graduate into outright Demonic Spiders.
Their saving grace is if you can wound them when they attempt to throw, they'll fumble and blow themselves up. There's even an achievement for it if you can cause a sapper to take out himself and at least one other mook with a stick of dynamite.
Idiosyncratic Ship Naming: Lara x Sam Les Yay ship has been dubbed S.S. Endurance (Same-Sex Endurance) by a portion of the fandom - most predominantly in Tumblr art
Internet Backdraft: Be warned about bringing up the Attempted Rape scene. Though this has abated somewhat now that the game is out, as the scene plays out somewhat differently. While the man was obviously planning on, or at least strongly considering, raping Lara, as soon as she fights back — which she does without player input — he immediately tries to kill her. Failing the quicktime event results in Lara being strangled to death.
Les Yay: If Lara and Sam were explicitly stated to be a couple or one was romantically interested in the other, absolutely nothing about their relationship would need to change.
They seem to be very couple-like during Roth's funeral, and their first scene (shown via recording) is straight out of dozens of romantic films, with the concerned Sam trying to get the workaholic Lara to take a break already. Sam even calls Lara "sweetie", and does so again in a later recording.
Sam being a Distressed Damsel is Lara's primary motivation for much of the game, is willing to go to extreme lengths in order to rescue her, and the final levels heavily resemble a Rescue Romance. Lara bridal carries Sam down the mountain after rescuing her, while Sam is wearing a white dress.
In the prelude comic, the last page is a splash image of Lara hugging Sam close with one arm with the caption "The Beginning."
Much of the in-universe Male Gaze comes from Sam's camera lingering on Lara's and Reyse's bodies.
Between this, Mirror's Edge, and Heavenly Sword, "close female relationships that can be interpreted as romance if you turn your head and squint" appears to be a Rhianna Pratchett trademark (and if you spot her Faith/Celeste).
The tie-in Comics to the Reboot have taken the Lara/Sam relationship Up to Eleven with Lara outright stating that she Loves Sam more then anyone else in the world, Sam calling Lara babe when she is talking to her. Scenes where they are together are acted out as if they are a couple with Sam calling Lara adorably awkward and Lara stating that she doesn't care what her enemies motives are as all she wants is Sam back. One scene even has Lara surrender to the villains, risking her life and the life of someone who had just helped saved her, because it was the only way to get to Sam.
There's also this◊ from Gail Simone, one of the writers of the tie-in comics.
Moe: The game's executive producer Ron Rosenberg explained that the Attempted Rape scene was meant to portray Lara as such that the player would want to "protect" her, apparently implying her to be this trope. The result was the Internet Backdraft, and the studio backpedalled on the statement.
Moral Event Horizon: Whitman's betrayal, handing Sam back to Mathias and trying to let everyone die just so he could have a good story to show the media. Needless to say, he pays for this big time.
Spiritual Licensee: Survivors on a creepy island which no one can leave because of weird weather, targeted by a cult of hostiles who either assimilate them or kill them and needing to go to a specific area so as to leave the island, this is pretty much Lost the game, only with a significantly tighter plot. Lara's outfit also resembles that of Kate in the series.
Squick: Some of the death scenes when a quicktime event is failed can be very gruesome. For instance, there is one where Lara gets shot through the head from the chin up, which leaves her with a giant hole in her forehead and an eye popped out of its socket.
The impaling deaths, from spikes to tree branches to swords, are always gruesome. The designers seemed to have taken a perverse delight in animating Lara twitching and gurgling for a few seconds before suddenly becoming very still.
How about going knee-deep in chopped human carcasses which are not a mere textures, but objects affected by that walking? Or taking a literal blood bath? The game is littered with hundreds of corpses, many of them mummified, many... a lot fresher. This is unquestionably one of the most horrifying things to ever appear in a video game.
That One Level: The river rapids and immediately following parachute trip. Camera Screw makes it incredibly hard to tell where the safe paths through the obstructions are, combined with controls (at least with mouse and keyboard on the PC) that aren't particularly responsive. Get used to seeing Lara dieover and over again.
The Woobie: One of the reasons that the game's dark tone works so well is that it's almost impossible to not want to hug Lara by the end of it. Or after her first kill.
Iron Woobie: But she doesn't back down once she's finally had enough and begins to fight back in-story.
Jerkass Woobie: Reyes would count due to her Jerkass attitude towards Lara could be result of not just Roth's death and Roth's favoritism towards Lara then her as Reyes was in a relationship with him, but also because of being estranged from her 14-year-old daughter and the loss of her job as a police officer. Roth's death only adds salt on the wound to Reyes' stress and eventually causes her to vent out her frustration of the culmination of her personal problems towards Lara throughout the game.
They Changed It, Now It Sucks: Some old time Tomb Raider fans have claimed this, due to the introduction of cover-based shooting and the reduced emphasis on puzzle solving compared to the early games.
They Wasted a Perfectly Good Character: The only characters (besides Lara) that get significant development are Roth and Sam. Reyes, Jonah, Whitman and Grim mainly only get enough to establish their personalities, that Reyes and Roth are in a relationship, Whitman's Face–Heel Turn is inevitable and Grim was also an old friend of Lara's father. Alex gets so little his crush on Lara comes completely out of nowhere in time for his Heroic Sacrifice.
What an Idiot: Lara has a few moments, being Genre Blind. E.g yelling over the radio to Sam, when Sam says she has to hide the radio. Of course, she does get better.
Even in final stages she still has her moments, like allowing Whitman and Mathias pass right next to her, having perfect opportunity to stop them. Or using bow during powerful blizzard while shooting against the wind - she is visibly surprised that it didn't work. In Lara's defense regarding Mathias and Whitman, there were Stormguard in the the area, alerting them to their collective presence might not have been the best idea.
Just before that final fight, she had an arrow pointed at Mathias while he was performing the ritual. No blizzard yet, Mathias was obviously distracted... all she had to do was release it and it would've been over. Instead, she just keeps watching until the ritual is fully underway and the blizzard whips up, making it that much harder to get to Mathias and Sam. If the ritual was instantaneous, it would've been over right there and then.
By the time that Lara runs off to find Sam at the end of the game, when asked, Lara says she doesn't know why the Solarii want Sam. Are you sure, Lara? There wasn't anything at all that gave any hint whatsoever as to why Sam, a descendant of Queen Himiko, might be considered important to the Sun Queen's "power-transferring" ritual? You didn't witness any events or hear any conversations or find any clues that would possibly give any indication as to why Sam was taken? Really? The fact she only figures it out right before the Point of No Return when most players would have pieces it together hours ago only further drives that feeling home.
When Lara points out Whitman's claim about being chased is rather shakey and makes reference to something he did previously, rather than have her explain what happened, which would lead to the obvious rationale that Whitman was planning on betraying them and probably made a deal with Mathias' people, Reyes and the others basically ignore this and instead force the two not to argue....thereby giving Whitman chance to hand Sam over to Mathias as soon as Lara's away.
After Lara's friend and mentor Roth's funeral, Lara is so broken up that she requires some time alone (fair enough), and tells the others to go on ahead without her, and nobody calls her out on it. Seriously? On an island crawling with dangerous animals and deranged cultists, which she has just fought through tooth and nail to find them again, and has already claimed the life of two noted badasses, she decides to split up the group? The result of this decision is nothing particularly bad, beyond another couple of battles she had to fight alone, but it COULD have been much worse, and they would have had only their own stupidity to blame.