Bands do get sick of their own songs, but most of the time it's a consequence of stylistic changes rather than playing the material all the time. Sometimes it's not, though; at DIY shows, you tend to get people yelling for stuff off a band's early records, and occasionally the band will respond with something on the order of "no, man, we're sick of playing that, let us do stuff off the new one".
Personally, I haven't gotten sick of any of the stuff I've written and recorded/programmed. Some of it sucks
, but getting disenchanted with your work is different from getting sick of it. The band from this video
, who I'm good friends with, got so sick of the song in it that they refused to play it live as soon as they could possibly justify its omission from their set, both because their style changed and because it was such a huge early hit for them. No idea about other bands' reactions to their big hits, but if they have any ambitions to be taken seriously as musicians rather than a jukebox that plays that one hit, it's likely to be similar.