Do beavers' tails hurt from being dragged?
No, it does not, otherwise beavers wouldn't walk with their tails dragging. Here is a picture of the bottom of a beaver's tail:

Found here
and here is a closeup (probably of the top, but they're very similar):

Royalty-free stock photo from Dreamstime
As you can see, the skin looks almost like reptilian scales. It is likely highly-keratinized and is quite thick, as you can see below. Innervation is probably net very close to the surface, like callouses we develop on our hands and feet, so it doesn't hurt to be dragged over sharp surfaces like rocks or thorns. Here is a cross-section of the thick part of the tail close to the body:

Source: Cannundrum blog