At around 57 minutes, after Lara escapes Mathias (as the scene turns to night), there is a close up of a Bearded Dragon in a tree. This is a desert lizard native to Australia and could not survive the humidity of the island.
When falling in the river Lara ends up on a rusty crashed airplane.
The body work of that plane would had been made of aluminum, as iron is way too heavy. This would not result in a brown rust patine, as aluminum turns white when it corrodes.
Just before the 'Fox Hunt' near the start of the film, a small hole is punctured in the can of paint before Lara sets off on her bike. Unless another hole was punctured in the top of the can as well to let air in, the paint wouldn't leak out of the can at all.
The goof items below may give away important plot points.
Lara takes many unnecessary risks in the final battle with Vogel. She approaches him from behind, and could easily have kicked him down the shaft that the makeshift bridge goes across. Instead, she jumps out onto the bridge, dangling off the side as the nearly falls into the pit, putting herself in a vulnerable position. She then takes the diseased finger and crams it down his throat, potentially risking getting herself infected. And on top of this, she kicks him after he's clearly infected, running the risks of infection herself once again. All of these risks could have been avoided if she'd simply surprise attacked Vogel and knocked him down the chasm.
When Lara sees her dad she can barely walk over to grab the rope and looks to be hurt bad but when she misses the rope she starts climbing the cliff face to get up to the cave. If she was hurt bad enough to barely walk she wouldn't be able to climb the cliff.