Hi, pkshima!
I had similar problems with other PIC projects (several), and I find this is caused by high frequency (auto-oscillations) produced internally on the PIC, mainlly in the ones what uses the internal oscilator instead crystals.
The solution I used to solve this: put resistors (arround 3K3 is OK) in the unused pin´s ports to GND!
This ends the strange behaviors you´re having. I noted too what this depends on the way the circuit is wired, and the PIC itself (some of then will do present the problem, others don´t!).
Anyway, I hope this can help.
Good lucky!