I take it back!

The problem with what I told you about grounding the tape is this: if you short-circuit the heater to the tape accidently, the current will run through the tape to ground.

Problem is, the tape is thin, so it could catch fire.

This is worst case scenario, but can theoretically happen. Therefore, do not hook the ground wire to the tape.

It may solve the noise problem, but is not wise in this application.
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Mark