There must be loads of these. I'm sure ER is not in fact a Bible to medical correctness (say it isn't so!).
The one that bugs me is the bit the West Wing where the President breaks out into Latin. It makes me cringe. It's bad, bad Latin (IMHO).
And also - of course - the fact Toby never gets a chance to clear his name. I love Richard Schiff for explaining he played it assuming Toby was covering for someone, which is great explanation of a frustrating storyline. But I never believed Bartlett (who seems to me so good at accepting people's different points of view) would have judged him so quickly.
Is there anything that bugs you massively? Either for being factually wrong or just not fitting your sense of the characters.