No it doesn't. I have many of my git repos in Dropbox but I'm not using Dropbox for sharing. Having those in Dropbox means I get automatic backup and that they are available when I switch to a different computer, which I do, but not frequently. As only I use my Dropbox account, I'm aware of the potential sync problem, but it's never been a problem. I do run fsck & gc more frequently than most, but I probably don't need to.

EDIT: I should emphasize that this model is way more convenient than manually having to remember to push and pull all the time. Now push is only for publishing outside as it should be.