There are some shows that have opening and closing music, of course. It helps with the credits, but if we were to include it, it would be for thematic reasons. I think that worked really well for the BBC Narnia mini-series:
I might be in favor of a musical closing if it were instrumental.
Most animes have opening and closing music, and typically these are J-Pop songs (though they can be other styles, depending on the type of show it is). Here's the closing music for Gunslinger Girl:
Basically, though, you have to be careful not to eat up too much of your screentime with opening and closing sequences. A typical one-hour TV show is already down to about 42 min after time for ad breaks is factored in. We're not following that model, so our episodes are one hour, no breaks. So, we should have time for, say a 30-45s closing musical sequence, if we wanted it. It would give the audience a chance to switch gears from being in the world of the show to coming out of it and being in the real world again.