Let's see. When I installed my DSL modem and a linksys router for the ethernet set I had to put filters on the other voice lines to filter out the dsl noise.
It follows then that you should be able to install a DSL modem on the second line. It couldn't really hurt anything to try.
But, given the choice, I would do like the others recommend and set up a wireless access point and put a card in the second unit, then the location of the second unit would not be limited to where the phone tap is.