Most of the time the SC2.exe process doesn't exit when I close SC2. I don't believe it's a problem with my hardware. I'm not sure what other details I can give. It seems to do this regardless of what I did when SC2 was open.
Is this a known bug? Do I need to give other details? If so, what details? My dxdiag info is posted in this post (still up to date): http://forums.battle.net/thread.html?topicId=23425556685&postId=234232599551&sid=5000#4
When it's hung, it's using about 60-70 MB of memory (so it has released a good deal of it, as it's more like 600+ MB in-game) and no CPU, and prevents another instance of SC2 from coming up. It also prevents my Microsoft Multimedia Keyboard's (handled by MS's IntelliType Pro software) special buttons from working (including mute, launching a browser, etc.).
It's been doing this since the start of the beta. I haven't found anything to make it do it more or less often.

Hello Tiger

Can you close out any other programs you have running prior to launching StarCraft II and see if the issues still occur? What often happens is another application will latch on to something in the program and when you close out the game the other program will keep parts of it open.

Also, you may want to give it a couple of minutes to see if it ever closes out. I know on my system Firefox tends to run for about 30 seconds after I close the last window and something similar might be going on.