I've learned the hard way that some painted surfaces have to be sanded first. If the finish is oil based or very shiny it will not hold the paint. I've not painted furniture and so I don't know about yours but I've had to strip and repaint an entire wall in one house and doors in another because the coat I put on pulled off the wall and door. Just a light scratch would reveal the paint underneath.
I was told by the person at the paint store that when the first coat is oil based and one tries to paint over it with water based paint (can't remember what it's called) the paint does not adhere to the first paint.
You can test this out by painting a small space, let it dry, and then scratch the surface to see what happens.
Or talk with the knowledgeable clerk at a paint store. BTW I've found staff at big box stores to be just as knowledgeable as those in specialty stores about general and basic things such as this.
If the wood is laminate, as most IKEA furniture seems to be, definitely do not try to strip it. The moisture will cause the laminate surface to buckle.