I submitted manuscript to a special issue of a Springer journal. The review date has passed. What should I do now?
I would not worry about this at this stage.  In publishing (and organizing conferences, etc.), almost everything takes longer than the original deadlines would suggest.  At this stage, the reviews may have been completed; but then the editors have to go through all the reviews themselves and decide what to do with each paper.  Moreover, there are probably a few referees who are late turning in their reports, and the whole process can be pushed back several weeks.  Add to this the fact that early December tends to be a very busy time for many academics (semester winding down, final grading to be done), three or four weeks from the reviewing deadline seems a totally unremarkable delay in hearing back from the editorial staff.

I would recommend not doing anything, until at least some time next year.  However, if you wish, you could send an e-mail to the editor, asking about the status of your submission.  Keep in mind though that replying to requests like that just take up more of an editor's time, and it will probably have very little marginal effect on how quickly your paper is dealt with.