I say no, not on the spud washer. Bolts, ok.........
Just installed new toilet . When you put the tank down on the toilet when you put the spud washer on should you put or spread some pipe joint on it just to be sure it doesn't leak. On the washers that hold the tank on they have rubber washers on the bolts, do they need compound on them also.
The Pipe dope won't seal in this case... it will lube so that the gaskets slide instead of bind..... The pipe dope won't hurt, but I'm not sure what it'll help... do what makes you feal good in this case. have fun with it.
I still wouldn't dope the spud washer because I've see the spud washer slip out of position and egg shape and that's not good. I wouldn't dope it.