Yes, you can get the paint off the woodwork, you just need to be careful with whatever product you pick to use.
Here are some suggestions:
-If its not oil base paint, You can remove the paint drips with a bucket of water and a wet rag. Wipe the wet rag across the floor and let it moisten the paint drops for about a minute. Then go back over it with the rag with a little rubbing. It will come up this way I have found the best without using chemicals. If its dried oil base, you will need a alcohol base cleaner to get it up.
-Try lightly sanding with a light grade sandpaper, but that would probably take the white off and then you would have to recover that area with white paint.
-Buy some paint remover at home depot or use finger nail polish
-Try goof-off, though some of the finish may come off with it. You may need to touch up the finish with polyurethane after getting the paint off.
-Go to the hardware store and get some denatured alcohol and put a dab on a rag and the paint will rub off without doing any harm to the hardwood floor.
-There's a cleaner that painters use called Krud Kutter.
I have also read something about putting olive oil on it and letting it sit for a bit and the paint will peel right off.