Can I lay tile over drywall that has had the paper torn from removing tile that was installed wrong when doing new home construction?
Given the cost of tile and the value (or cost) of labor, I'd go with ripping it back to the studs and installing a proper tile backer-board (Durock®, Hardi-Backer® or the like) rather than fuss with a dubious substrate. It will be faster than trying to repair the drywall surface and it's a better surface for attaching tile to.
Typically, no, if you're using a mortar-based mastic. Proper adhesion depends on the mastic drying in a controlled, relatively slow fashion. Drywall with torn paper is more permeable, and mastic dry times will be inconsistent.
However, this doesn't mean you have to rip out the drywall and start from the studs. You can apply a skim coat of drywall compound, sand it down, apply a second, and sand again. If you still see puckering from the raw paper behind, add another coat. Lather, rinse, repeat.