Well, I've done this a few times when moving into cheap apartments back when folks smoked in their homes and exhaust fans were the work of the devil.
The pithy answer: mild degreaser, plastic scraper, lots and lots of elbow grease. The idea is you want the gunk to soften up and then help it peel off. Then deal with the stubborn parts that will just take more work.
Depending on the finish some might be pulled off. If the finish is cloudy or crazed in any way, it'll come off pretty much with any aggressive activity.
Per the comment, maybe try a mild degreaser in a hidden area to see how effective it is. If you need to step up to a strong alkali degreaser to really render that gunk, you can try the same area and make sure the finish can take it.
This way you do it in steps, from the minimum to "oh geez, gotta refinish it anyway". Which is always a possibility when refurbishing old stuff.