The only time I have another tab open while playing a game is if I'm playing Shell Shock Live with a friend from this site. A multitude of problems can arise while playing that game, and it's good to have that user's and my own tabs open so I can easily communicate.
Otherwise, I don't find the Tavern important enough to interrupt a game.
I sometimes switch back and forth, depending on which game I'm playing. If the game is entertaining (or my opinion of intense) I continue to play. If it is boring or slow, I don't mind checking any of the forums and post something.