How can I remove white 'splatter' which has partially seeped into a wooden panel?
Looks like wall paint marks from a roller getting too close as the wall was painted, but there's no easy way to know if it is oil based on latex based paint. Either way now that it is dry and old, it's going to take some sort of solvent to remove it and that will also remove whatever finish is there as well. 

You really don't have any options that I am aware of to avoid having to refinish it, I would just plan on doing that. 

One possible option is to try using some nail polish remover, because it is decidedly mild since it is intended to make contact with skin. That might soften it up enough to make it come off without doing too much damage to the original finish. Try applying it very sparingly with cotton swabs.