What kind of paint and what kind of finish?

Latex paint will often come up with denatured alcohol or xylene with minimal damage to the underlying finish unless it is shellac.

Alkyd (oil paint) is harder.
Regular paint thinner can often soften it enough for removal.

Any dots of paint should come off by lifting with a single-edge razor blade. Tiny atomized spatters or smears will come off with some 600 grit wet/dry sandpaper or a rubber sanding block, lubricated with paint thinner or water+detergent. If the old varnish is nice and thick you don't have to worry about sanding through the finish to bare wood. Pick off the big flecks, then abrade the mist/smears.
Casey