The best time to remove paint splatters and spills is when they're fresh. If the paint is water-based, wipe up as much as you can with a paper towel or a rag, and -- before it has a chance to dry -- wipe up the residue with a damp rag. The procedure for oil-based paint is basically the same, but moisten a rag with mineral spirits instead of water to wipe off the residue. Mineral spirits is harmless to most furniture finishes, as long as you don't rub too hard. Avoid leaving water or mineral spirits standing on a clear finish, or the finish may turn dull.
Once a skin has developed, you'll probably find it difficult to remove either latex or oil-based paint just by wiping, but there are ways to increase your chances. Spray the spot with spray lubricant, which won't harm the finish and loosens the paint adhesion; then scrape the surface with a plastic paint scraper. A plastic scraper won't harm the finish, and it has enough rigidity to get rid of one- or two-day-old paint. Wash with soap and water or mineral spirits to remove residue that has settled in the grain of the wood.
A small problem can turn into a larger one if you scrape or wipe a spill from a varnished surface and don't wash off the residue immediately. Once that thin coat of paint has dried, it's very difficult to remove it without affecting the varnished surface. If the paint is latex, wipe the reside with denatured alcohol, which will emulsify and remove the paint, but if the finish is shellac or lacquer, expect some damage to the finish. Lacquer thinner will remove oil-paint residue, but it's virtually guaranteed to affect the finish, so use it sparingly, or, as an alternative, sand off the residue with 220-grit sandpaper.
Paint spills that have been allowed to dry and cure are the most difficult to remove without damaging the finish. Loosen the bond of hardened oil- and water-based paint on its substrate by spraying it with spray lubricant; then, after about 10 minutes, follow this by scraping with a razor scraper. Keep the blade of the scraper low with respect to the surface you're scraping, using gentle pressure and scrape with the grain of the wood. You may be able to avoid damaging the finish, but if not, sand the damaged area with 220-grit sandpaper and touch up with clear finish.