Yes, ironing seams can be done, however not in this case, as you've painted the wood; so do not attempt using an iron or you'll have a mess.
The water-based paint softened the glue of the veneer, and the wood absorbed water and curled...as wood will often do.
Using a hypodermic, inject a good carpenter's glue in under the edges that have curled. Then lay wax paper or similar over the face and weight it down. After the glue has thoroughly dried, sand the remaining raised edge and seal the exposed wood with a coat of lacquer or urethane before repainting. When you repaint, try and do it in several thin coats, so that the paint dries quickly and doesnt cause new problems. Good Luck