The only way to get the finish off is chemicals...unless you want to sand it off....which would ruin the piece.
Post a close up of the damage....you may be able to fix it without a whole re-do.

It depends on the current finish. If you have an oil finish, you may not need to use chemicals to strip it. If it's a veneer, you want to be careful about sanding. Go to youtube and watch some videos about refinishing furniture --- but figure out what finish you have current first.

That is a beautiful set. I used to drool over the HW furniture at a store that specialized in them.

It looks like the chairs are in good shape in terms of finish and it is the surface of the table that needs to be finished. Honestly I would get a professional to do it as they would be able to match the finish to the rest of the set. I wouldn't start learning on a good item of wood.

I had a Waterfall set with zebra veneers that was in rough shape after my hard use. I had the set professionally refinished and it looks beautiful.

My ex-husband and I owned a Heywood Wakefield DR set from the 1940s. My ex kept the set in the divorce. It was actually left at the first cabin his family had purchased, as I'm guessing the people who owned the cabin before didn't know what the set was really worth. HW furniture is solid wood and won't have any veneers. I found a site below that claims to sell kits that replicate the original HW stain colors. Yours looks closer to the Champagne shade.

I would probably try to find a small inexpensive piece at a yard sale or thrift store to practice on, and then try my hand at the dining set.

Good luck. Come back and let us know how you make out

https://werefinish.net/

Sable is different than the Champagne. I have two sable pieces. The pieces I have are from the Cadence line and I have read they were made betw 1955-1959. One of mine is dated 1957.

I am curious of how you did with the table?