Today I accidentally plugged my cheap master Ethernet test unit into my brand new router which had passive PoE enabled on the port. The tester went crazy and now skips pins. It will be cheap to replace but could this potentially hurt the router itself?
It shouldn't, but you can't know for sure after you've put the router under test.
It depends on the proficiency of the electrical engineer (or commercial limitations) designing the power supply circuit in your particular router model: Judging from the usual implementation of this non-standard PoE implementation, the only real danger here would is the tester shorting out the power pins.¹
Shorting out results in a huge current draw, against which any decent power supply (circuit) must have protections, as the potential damage of overloading the supply may not only destroy the circuit itself, but may result in fires.
The protection can be designed in many different ways, but in the end it comes down to wether this protection is resettable, or - literally - blows a fuse. In this case, you can no longer use PoE on the Port, but it should otherwise function fine.
¹) Assuming an (otherwise) standards-compliant implementation, the other pins are electrically isolated from the power supply circuit. See the EE Site.