Without mentioning names, some users seem to just post entire tweets looking for translations, and I think there are some issues here.
First, I am not against tweets as a source of study. It can be good source of natural, commonly occurring Japanese, which is exactly what we want.
However, if a tweet is to be posted, I think we should first insist that it be cleaned of all irrelevant emoticons and extraneous text to get at just the text needed for the question. Also, that it necessarily post a link to the original tweet for anyone who cares enough to try and get the context.
Is there a way we can make that requirement written into the wiki or a rule?
As a side issue, I think a lot of tweets represent highly personal communication within in-groups, and sometimes use terms that are almost entirely exclusive to that one interaction.
Having thought about it a lot, I don't think there's any good way to restrict what kinds of tweets get used for questions. However, I think it is fair to downvote a question if it contains such unique Japanese. My thinking is that understanding Japanese of that type is the result of having got to a certain level of Japanese, it's not the route to that level.
Just thought I'd put that out there.