There is no way to close a case that you had opened in App Studio. However, in my tests, I noticed that old case threads are automatically removed from the clipboard and replaced by newer case threads. So there is some "cleaning" going on behind the scenes.
Although I had seen this issue before, and if I tried to deliberately replicate the issue, I wasn't able to. I guess a way of resolving this is to logout of App Studio, and login again to get a clean slate.