Welcome to the Forum from New Zealand.
It sound like you are from the USA with a 2.5L Z3. Hope it's not too cold where you are.
P1620 say that the heating element in the thermostat is faulty. The thermostat will still be opening and closing as it should. The heating element is controlled by the DME and is used to force the thermostat open more when the DME sees increasing load. Think of it as pre-emptive cooling control. You will need to replace the thermostat at some stage, but its not that important. Get it done before the summer heat.
Only cold air from the heater has 2 lines of diagnosis:
- Blend flap control
- Water circulation
The Z3 heater air control has a blend flap that has a habit of sticking in the cold position. When you turn the temp knob, the little nub on the back of the wheel breaks. If the air temp knob turns freely with little resistance, then this has happened. Do a search for "Blend Flap" and you'll see how people have replaced the nub with various items and adjusted the bowden cable to the flap to stop the flap from sticking again.
Water circulation includes:
- coolant level/entrapped air
- Heater valve
- Heater valve micro switch
If you have no heat when stationary but do have heat when moving, then this is a sign of entrapped air in the cooling system. Bleed the cooling system.
The heater valve can block up. Sometimes it is the rubber valve itself that has died with age and blocks the valve. The valve is on the fire wall, behind the end of the inlet manifold. Time to get your multi-meter and wiring diagrams out to check for voltage at the valve. Do not expect 12V when testing at the valve. I don't know why, but my testing of a working valve only showed something like 3 V.
The micro switch in on the back of the temp control panel. It is set up to change state when the temp knob moves off the fully cold setting. Again, this is a multi-meter and wiring diagram job.
You can find the wiring diagrams in the "Resources" section of this forum. See the blue ribbon at the top of your screen. Use the build date of your Z3 to select the correct set of diagrams.