Can cable ties withstand high temperatures?

They should withstand boiling temperatures just fine. If they are touching a part of the exhaust system near the motor they may melt. 

Heater. Where?

The heater Core on your car? Use the black ones from a hardware store, not the Harbor Freight. The coolant  in them doesn't get above 192, plastics Start to melt at 250. Many New car heaters are partially Made of plastic..my '15 M 3  has Plastic hose clamps in places. The black ties are made for outdoor use, much more rugged than the clear ones. Clear R  OK for interior wiring looms, harnesses.

A Space Heater? Put  No flammables near a heater element.

MOST plastic cable ties are not safe for high temperatures.

The expensive nylon ones I use will take the heat. The cheap Chinese plastic ones will give way.

Most cannot - no.

You should cut the loose end off - otherwise you risk starting a fire.

What "heater"? The heater assembly in your car is all made of plastic and is not "high temperature". 

Which heater? 

What is the temperature of your heater?

Not really they will either melt or dry out and snap.Use metal wire or strapping.