On RPG.SE this shirt had a very perceptible result: we had a lot more downvotes for questions that were getting closed, when normally we have one or none.
RPG.SE's community members are usually pretty generous with their upvotes, and careful and precise like a surgeon with their downvotes. They rarely feel the need to downvote a question that's going to get closed, instead leaving it at 0 or -1, or lower if it actually didn't do research effort, is unclear (e.g. totally unintelligible), or isn't useful. Our downvoting and close-voting are very much disconnected.
In the first half of Winter Bash, things were different in the way you might expect: all the questions destined for closure were getting downvoted to -3, or past -4 and taken right off the front page. This was regardless of whether the question was well-researched, useful, and clear — it seems some users were just slipping in a vote to get closer to their Red Shirt hat. (Hat-shirt? Shirt-hat?)
Not sure if this is a good thing or not, but that happened. I'd wager it's not a good thing, since we shouldn't encourage people to treat downvoting and close voting as things that need to happen together.
(That said: I suspect this specific hat was a social experiment on behalf of the SE staff, and I'm glad it happened, and doing it as part of Winter Bash with a hat was probably a very good way to execute it. If it wasn't deliberate, it was a pretty good accidental experiment to conduct!)