Here are a few from memory.
A Bit of Fry and Laurie: One of the sketches had some music (badly) replaced due to licensing issues. The Netflix version had the original music intact, but I'm not sure if the show is up on Netflix anymore.
Monty Python's Flying Circus: Graham Chapman sings a couple of lines from a song, which has been cut on every DVD AFAIK (don't remember which series).
The Wrong Trousers: Music has been changed on the DVD and Blu-ray releases, including 'Happy Birthday' and 'How Much is that Doggie in the Window'. The original music can be found on VHS and Laserdisc releases.
The Peter Serafinowicz Show: One sketch was cut from the DVD release due to licensing issues. The sketch can be seen here
The Young Ones: A whole bunch of music has been replaced or cut across both series on DVD.
Fist of Fun series 2: Not music related, but the BBC, for whatever reason, insisted that some jokes about the royal family be cut.
There are of course many more, but these are the ones I could think of off the top of my head.