Need to take down a wooden wall-mounted coat rack, but screws are sunk into the wood

If you don't want to damage the wood have you considered asking a tradesman to take it off for you? I know it would be a quick job for someone with experience of taking out difficult screws like myself, it's a £10 job at most and could save you the heartache of damaging the coat rack. Hope this helps

Have you tried an impact driver? Make sure to apply as much lateral pressure onto the screw as possible. Or you will shred the screw head. It also wouldn’t be impossible to wallpaper around the coat hanger if you’re careful with a brush and knife when trimming it.

Try putting tension on the screws by slipping a very thin scraper or filling knife behind the rack. Then when tension is applied as you unscrew it it should come off the wall fixing. To remove the screws buried in the timber baseplate I would support the hole in the timber either side of the hole with a finger each side before tapping it gently out or unscrewing the screws.

try using a large hacksaw blade by itself not in the hacksaw,gently sliding it between the wall and the coat rack, you could also slide some card on the coat rack side of the blade this protecting the wood, the slowly cut through the screws.

It sounds as if you've already attempted to remove the screws which means the heads will have already have been compromised. The best advice I can give is to get an experienced contractor to remove them for you. There are various methods to do this ranging from screw grab products to using an easy out which requires drilling into the head of the screw with a smaller diameter drill bit and hammering the easy out into the drilled hole. It has a reverse thread so ratcheting it in removal allows it to grab and remove the screw. Really easy and quick, an hours labour at the most. Hope this helps.