Add an LED to a low power line (heater thermostat switch)
Could it put some impedance load on the AC line or disturb it somehow?

If you wire it as a current transformer (i.e. a single turn in the primary) then it should be OK but, you do need to know the full load current of the heater to calculate the number of secondary turns.
So, if the heater current is 10 amps and you wind a thousand turns on the secondary, it will drive 10 mA into an LED. But you need a reverse diode across the LED or maybe another LED wired in reverse to the first one. You don't need a capacitor or series resistor for this to work but, you do need to provide sufficient secondary turns so that the LED doesn't get fed too many mA: -