Here is an excerpt from Eric S. Raymond's HTML Hell Page:
Blinking text makes it nearly impossible to pay attention to anything else on the page. It reduces 87% of all surfers to a helpless state of fixated brain-lock, much like that of a rabbit caught in the headlights of an oncoming semi. This is not good. If you abuse the blink tag, you deserve to be shot. Clue: if you use the blink tag, you're abusing it.
This applies to animated GIFs and flash. ESR's page goes on to say this about animations:
With animations you get the all the wonderful injuries of the blink tag with the added insult of the graphics download time. People who abuse these should have flip books rammed into every body orifice until they figure out that a two- or three-frame graphics loop is even less pleasant than that.
My biggest irritant in this category are commercial web pages that offer a free service, but pay for it with advertisements. Some are tolerable (a few search engines, such as Google) but companies (such as the now dead deja.com) that cram several animated gifs into a single page are really overdoing it. I understand that advertising "pays the freight," and these companies get more money from a click, so naturally they want to draw attention to the ads. However, overdoing it is just going to drive people away in the long term.