The questions a

1) Why would a brand new furnace be leaking power? And how's it
getting into the shower? The only other electrical devices in the
building are lights, and a couple electric sockets.

Through the air. AC at 60hz acts like a radio wave in that it can
induce a voltage in a conductor near by. Any wiring, motor or
transformer will radiate electromagnetic radio waves. The plumbing
is the antenna that is being induced.

2) Why isn't the power going out the neutral and ground?

The felt voltage is not coming directly from the wiring.

The shower head is the hot lead of the voltage source and the wet
ground or drain is the other conductor. Electricity has to have a
path to be felt.

3) How to safely take care of the problem so the campers aren't
being shocked?

Short the two leads. The shower head and the floor drain.