We had a mouse underneath our kitchen sink who ate through one side of the wire from the dishwasher power line. The wire is exposed, but again just one of them. I've unplugged the dishwasher for now. To pull out the dishwasher and replace the whole power line would be a big undertaking. Is it safe to just cover the exposed line with electrical tape? Or do we need to replace the whole thing?

If it is back in an area where it won't be disturbed much, I would do it. You might even reroute it keeping it back against the wall even nailing it in place with staples meant to hold electrical wires. Even less of a problem if it was the white neutral.

Sounds as if the mouse ate through a cord, which plugs in for the dishwasher, since you say you have unplugged this.

Replacing this cord does not require the DW to be removed.There is a smaller finish panel on the front, below the door. Below that is the kickplate. Find the screws that hold on the kickplate to the DW, and sometimes there are screws holding the finish panel onto the kickplate or the frame of the DW. Once you have these two plates off you will see the wiring junction box at the front underneath the DW. Usually one screw removes the cover and you will find the cord splcied onto the machines black and white input wires. The ground may splice onto a green wire, or there may be a green screw insdie the junction box.

Taping the damage should be OK for a temporary fix, but replacing the cord is more appropriate. Be sure to use #12-3 cord.