"Forklifts" do not, of themselves, have unusual fuel requirements and special-purpose "Forklift fuel" is not marketed.
Diesel engines, operate on Diesel fuel, regardless of the type of equipment they are installed in. This is almost always identified as "#-2 Diesel fuel," however, "#-1 Diesel fuel" (AKA 'kerosene') or "winterized Diesel fuel," containing anti-gelling additives, may be used in cold climates. Some vegetable derived oils (biodiesel) or blends of Bio & petroleum Diesel may be used in Diesel engines where manufacturers and climate conditions allow. - 'Reprocessed' cooking oil is sometimes used in modified Diesel engines. (Don't just dump it into your Forklift or other Diesel engine without knowing what you're doing.)
"Off-road Diesel fuel" is #-2 Diesel fuel, with red dye added, sold without added highway taxes, for use in equipment that is not operated on public roads (otherwise it is the same product). You can use it in your forklift.
Yes, Forklifts, that operate on gasoline (petrol) and LPG are common. These units are equipped with spark-ignition engines, NOT Diesel engines (don't put Diesel fuel into them).
