I can usually shut off that part of my brain when playing video games. Only two times has it refused to be silent. Well, three times, but two of those times were in one game.
The first time was Borderlands 2. Okay, so a certain character was just murdered in front of me. Sad and all, but uh...why doesn't said character just revive? I mean, we have that machine that brings you back to life. We've been using it over and over throughout the whole game. And don't tell me said character can't, because it wasn't a problem before.
The other two times come from Uncharted 3. When your British partner guy pretends to be brainwashed, and then turns around and shoots his controller in the chest, killing him. Only, dude isn't dead. He shows up a bit later totally fine. The British guy even comments on this, saying, "How is that possible? I shot him!" The game never explains. Yes, I've heard that the developers said the guy was wearing a bullet proof vest, but nowhere is that mentioned in the game. It just looked like magic, and was a total red herring for me.
The other time is when Drake is stumbling through the desert, dying of thirst. He can barely walk, barely stand, and he is on the verge of death's door. You spend a fair amount of time just having him stumble through an ocean of sand. Finally, he falls into the ruins of a town, finds some water but can't drink it ("It's completely undrinkable"), and then falls through a wall into a mess of enemies. And suddenly, Drake is moving and rolling like he's perfectly fine. I just couldn't accept that. He should have been stumbling from cover to cover. He should have barely been able to hold his gun. He probably should have just surrendered, but we know that couldn't happen, so he should have ran. To this day, that entire scene still bugs me.
Honorable mention to Lara from Tomb Raider. She gets impaled at the start of the game--a wound that would slow anyone down--but after sealing the wound, she scampers around the island for hours, climbing walls and jumping off cliffs like nothing is wrong, and only remembers that gaping stomach wound toward the end of the game when the plot demands it.