Outside of it being difficult to do, there are more reasons this wouldn't work out well.
Correct me if I'm wrong, but does the site not allow for fluid timeline? As in, you can have two threads at the same time that are in different points in time, such as a thread in one of your characters' early history, then one in their very late history. Adding seasonal changes would severely limit that and make some things difficult to explain, like for instance if you had a very long ongoing thread. Don't end it fast enough and it goes something like this; "It's summer," "OH GOD NEVERMIND IT'S SUDDENLY WINTER." Plus, what if someone wants to have a winter or snow-based thread when the site's season says summer? That's kinda messed up, they have to wait half a year for a snowday thread?