Why (if at all) is finish paint necessary on interior wood surfaces?
Unfinished wood is very porous, so it's more subject to dimensional changes due to humidity. Also, it's very receptive to dirt and oil from skin and cleaning. 

I the case of a door top, it's not critical. I've done the same myself. You could simply apply a coat of stain or paint, which will improve appearance and partially seal the wood. A coat of urethane will not substantially increase the dimensions of the door either.