It really really bothers me when a character puts something down, and then walks away without picking it up, especially if they show them with it again later.
Something not so small that bothers me is when a victim is running from a bad guy or monster and then happens to knock them down, like with a baseball bat or something, and then they just take off running again. Fucking finish the job, you dumb ass! Hit him a few more times and he won't catch up to you again in 30 seconds when you unsurprisingly trip over your own feet.
Similarly when they walk in the house but don't shut the front door again, or open the fridge but never close it. I'm like waiting the whole scene to get back to that and missed the entire dialogue.
Just watched an episode of Star Trek: Strange New Worlds. In the episode, the ship's security officer, La'An, enters the bedroom of Khan Noonien-Singh when he is a small child. Proceeds to put a loaded gun down on his desk, have a conversation, then leaves the room. You're the chief security officer, and you just left a loaded gun in a child's bedroom!