A float tank does a couple of things.
Some jurisdictions require their use as a back flow preventer.
Some use it as a resivoir to run the by-pass line to allow for a SLIGHTLY longer time running without getting too hot for the packings and seals.
Some use it to provide a vacuum in order to allow for high pressure soap injection through the pump and coil (if equipped). This will only work with low speed belt-drive pumps that give ~7" of vacuum; high speed direct-drive pumps with only 1/2" of vacuum do not work.
There are potential problems with running soap through the pump and coil; check with your local professional pressure washer dealer for details.
Black smoke indicates an air starvation condition causing the burner to run too rich. Open the air band/shutter and check the coil condition. Look for excessive soot build up that could also be preventing air flow and possibly clean the coil.
White smoke indicates the opposite condition requiring you to close the air band/shutter.
Also be aware there may be unburnt fuel in the insulation in either case. A local professional pressure washer dealer will be able to assist you.
Hope this helps? Cheers.