How can I remove old paint from wooden door frames?
Dried paint does not come of with a wet sponge, even if it is water based. There is a plasticizer in the paint which hardens as it dries and loses it water solubility.

There are several techniques, and which one will work depends on many factors, such as the finish on the door frames before painting, the cleanliness of those frames, the type and age of the paint, etc.


Try using a plastic scraper - use the edge of a credit card (not an active one) held almost perpendicular to the molding and scrape downward. You can also use a plastic putty knife, but that edge is usually not sharp enough.
Use a metal scraper - harder to do without damaging the underlying finish, but if done very gently and carefully, it may get most off. There are specialty scrapers called cabinet scrapers that are especially suited to careful work like this.
Follow up the above two methods with a utility knife (a razor blade-like knife). Use it sideways as a scraper just on the painted area.
Use a pot scrubbing pad - (but this involves wet hands that you don't like; try latex gloves) use a cleaning solution. Mask the adjoining wall first to avoid scraping paint that you want to keep.
Sanding - if all of the above fail, you can try a sanding block or sponge.  This will definitely get it off but will cut through the finish n the molding mandating at least a partial refinishing. This may be the only technique that gets out the fine lines of white in the grain of the wood as shown in your pictures, unless you want to try to carefully dig it out with an xacto-type knife (a very narrow craft knife).


For all of these techniques, the molding finish may be damaged. If so, you can touch it up with a polyurethane (assuming the color in the damaged areas is not too different from the finished areas). You might try a rub-on poly if the gloss is close to the existing one. It is more forgiving in application.