Error 13 (E13) of HLN diesel air heaters relates to combustion disruption, and sometimes it will have white smoke coming out of the exhaust port.

Different issues can cause white smoke coming out of the heater exhaust port. It can come from combustion air intake or exhaust blockage. If you remove both the air intake and exhaust tubes, then can start the heater successfully, the issues would be from either the air intake or exhaust. Check if they are blocked. 

The white smoke can also come from a flooded combustion chamber. If so, you can first unplug the fuel pump wire, start the heater for a couple of times to clear up the extra fuel inside. (Don't worry about the error code) Then plug back the fuel pump and start. It should go back to normal.

If there is continuous large amount of white smoke from exhaust, it would be carbon build up issue that needs cleaning up the carbon in the combustion chamber. 

If there is only a small puff of smoke, it would be fuel supply issue - is the fuel grade proper for the temperature? Is there waxing along the fuel line? 

The above would be the most common causes of white smoke coming from the exhaust port, and you should be able to narrow down the causes after the previous procedures are performed.