the main concern with any children funriture is flaking and the possibility of lead paint. If you are new to the baby world, you should read up on the problems lead paint can cause. Lead based paint was a common product years ago and could be on old furniture. If this is an older piece get it checked before you do anything.
Newer furniture doesn't have lead, but it can still flake and peel with normal wear and tear. The crib we used years ago had a baked on finish that would not let other paints adhere to it without stripping first. My first concern would be to know that the paint would withstand the test of children and not flake off. The safest bet would be to strip off any of the existing finish.
You could test a small test piece on the back or underneath side to see if the new paint adheres to it.