At the refrigeration and air conditioning school I attend, there is a Toyota forklift that we sometimes use for moving equipment around. When I opened the "hood" (remove the propane tank, pull a lever near the floor and the whole panel with the driver's seat on top tips backwards, allowing access to the engine compartment) I found the engine was a straight-6 "3F" engine also used in the 60 series Landcruiser other than the fact it had a propane carburetor.
A brief search found that the same forklift as a diesel version may have had a "H" engine. Smaller model forklifts I found out used the "Y" engine that my Uncle's old Toyota Van also had as a gas/propane model or perhaps a "L" or "B" if they ran on diesel.
I looked around for a while but there was no mention of the "legendary 22R" in the Toyota pickup used to power a forklift even though the 20R or 22R engines were widely rumored to be originally designed for forklifts.
There is also the mysterious "1DZII" diesel which I believe is an exclusive industrial/forklift engine.