When you burn a CD to Mp3 format then back to CD does the sound quality go down?

Yes there is a loss of quality

In short form : Yes it does go down.

MP3 is a form of "lossy" encoding. When you 'rip' a song from a CD to mp3, you are actually encoding a .wav file to .mp3 file. Higher the mp3 bitrate, lower the loss of quality but there is a loss. This is why the mp3 tracks occupy much less space than the wav audio tracks.

When you are converting back from MP3 to CD (to a .wav file basically), your source is the lossy mp3 track, which you can not restore back to the original. Converting to a .wav file will not lose anything from the quality of mp3 recording but mp3 was lossy to start with.