If I were to raise an undead zombie but I wanted a skeleton instead could I burn or peel off the flesh of the zombie in order to obtain a skeleton? If I did that would the zombie's stats change to a skeleton's or remain a zombie's?
Yes, but you'll have to kill it and strip the flesh off first. Not necessarily in that order. Peeling the flesh off of a zombie would cause it to fall apart and die.
Animate Dead (PHB p.212) describes that you need to cast the spell on "a pile of bones or a corpse of a Medium or Small creature". If you pick the former you'll get a skeleton. Zombies have bones, but given how the rules phrase it you'll need to have a pile of bones to make a skeleton rather than pulling them out of a creature. So to turn a zombie into a skeleton you'll need to kill it, strip its flesh off, pile the bones, then cast Animate Dead on it. Voila, you now have a skeleton instead of a zombie!
I believe that you cannot, as Animate Dead ruling (PHB p.212) stipulates that you need to "Choose a pile of bones or a corpse of a Medium or Small humanoid within range". Using the stat block, as proposed by Jeremy Crawford's tweet (see below), the animated corpse becomes a zombie, a undead type creature. It ceases to be a humanoid and won't ever be a legal for the animate dead spell again.
I believe the answer above is incorrect.
Q: When animate dead is cast on a corpse/skeleton, do you use the Monster manual statistics, or add a template? — @Johnathan_Lin on Feb 8, 2016
A: When you cast animate dead, use the skeleton/zombie stat block from the Monster Manual or appendix D in the PH. — @JeremyECrawford on Feb 8, 2016