Usually, these old school desks were made of oak and if it had a finish, it was often worn off allowing the paint to penetrate into the pores of the wood. It sounds like someone has given up on the stripping which requires perseverance. The stripper will not hurt the wood and in fact the best way to strip this item is by using a flow-over process used by most professionals which will keep the surface constantly wet with stripper while the paint is being brushed away. The rough brushing is particularly important with oak because of the large pores which hold the paint. I didn't say it was easy.