I'd think about varnish on top and oiled, black milk paint elsewhere. If the varnish didn't work out, I'd paint the tops black, too.

I'd ignore the cigarette burn except to sand the edges flat and to try to fill it with varnish. Embrace that '50s Betty Davis style. Someone with a good eye and steady hands could probably obscure it a lot with artist's paints.

It's really a blank canvas at this point -- do with it what you want to end up with.

If you are going to paint: only recommendations are:
- Clean thoroughly with both dish detergent and water, then mineral spirits.
- Sand only enough to dull the finish
- Fill the burn with putty and sand to level
- Apply shellac or shellac-based primer (e.g., BIN) as a barrier / bonding coat. This will seal in any potential silicone oil (Pledge) contamination
- Paint