I think some songs would sound incorrect if you didn't try to at least give an impression of the accent. For example, trying to sing a Scottish traditional song or American blues song in "Received Pronunciation" might appease the PC brigade, but it would just sound stupid. If a comic effect is what you're after, then fine, but if you're trying to give a good performance and do the song justice, I think you have to try to sing it in the 'proper' accent.

I think it comes down to taking the trouble to learn the accent properly - if you do this, you won't go wrong. If you don't do your homework first, you risk annoying people who do know the accent properly (or at least, getting laughs you don't want).