Yes you can apply a clear wood finish over paint for added protection, but with stipulations. Almost any clear wood finish can be used except lacquers. The solvent in lacquer will damage most paints and possibly cause bubbles to form; this is especially true for water based paints.
The types of wood finishes I recommend are marine spar varnish, oil base polyurethane, water born polyurethane and acrylic urethane. The water base products will be easier to work with and dry fast. While the oil based products, especially marine spar varnish, will stand up to harsh weather better. If a painted item will be used outside make sure the finish is rated for exterior use.
You have several sheens to choose from; this includes satin, semi-gloss and gloss. The varnish has a high sheen even the satin, while most water based products will have a more true sheen.
Before applying any clear wood finish over paint I recommend wiping the surface with a liquid sander surface prepping solution. This will dull the sheen of the paint and allow the clear finish to stick better. Sanding before the application shouldn't be necessary.
I was wondering the same thing. I wanted to add varnish over a skateboard, some of the wood have a painted graphic over it. So I done some testing.
-Polyurethane: works fine, but you have to protect the paint job with a "Paint clear coat" before…but wont work over chrome or gold paints, and it has a yellowish tint.
-Marine Spar Varnish: have not worked for me. It make bubbles in the paint, even if it is protected by a "paint clear coat".
-Water Base Varnish: works on everything, even chrome and gold paint that says not to clear coat :). I just have to let it dry to know if it will peel off or make the paint yellow over time. The down side is, "I think", it is the less durable varnish of the 3. The good side is you can clean your stuff with water and it does not smell much.
Next time I'll try: "epoxy resin".