Now you know why varnish is rarely used anymore. If it is pure varnish/shellac, then damage to the wood is likely. Before sanding and refinishing, the nuclear option is to try to rub it out with denatured alcohol. The alcohol will dissolve the varnish and redistribute or remove it. You can then touch up the varnish carefully.
If it is lacquer, then use lacquer thinner. Remember, varnish and lacquer are not the same.
This is somewhat successful in some stains.