The biggest issues in regard to fuels and forklifts are as follows:
The engine type: Diesel or spark plug ignition
The computerisation of the fuel system: Are O2 sensors used to adjust fuel air mix for example
The exhaust emission system: Does it have a catalytic converter? Does it have a particle filter?
The consistency of the fuel: Is composition controlled by tight standards (LPG for forklifts in Australia allows a significant variation in propane/ butane ratios); and
How the forklift is being used.

Testing I have been involved with show diesel engines in forklifts produce the most consistent low levels of pollutants. And very low sulphur diesel fuels assist markedly in achieving low levels of pollutants.

Spark ignition engines reliant on catalytic converters are a problem wherever usage is low because the converter is rarely at its operating temperature. Pollutant levels were high until the catalytic converter reached its operating temperature.

And LPG spark ignition engines without O2 sensors, in spite of a reputation for clean exhaust were the worst, especially in regard to CO because the LPG gas composition varied so that combustion was incomplete with some mixes. CO levels were measured as high as 9% (though this figure is subject to question because it exceeded the accurate range of the emission level testing unit).

In respect of total CO2 production diesel engines have the lowest levels because of their high efficiency in converting fuel to useful work. Engines specifically designed to give maximum efficiency with petrol, and engines specifically designed to give maximum efficiency with LPG (can't be run on petrol because of the higher compression ration the LPG octane rating allows) are next, and dual fuel petrol/ LPG generate the most CO2 because of the low conversion efficiency in turning LPG fuel into useful work.

In respect of some of the more health damaging pollutants however, unless a diesel engine has a facility for dealing with these with special catalytic converters and/ or particulant filters, an argument can be put that from a health perspective these engines are not ideal.

Finally in respect of alternative fuels, the issues revolve around the composition and consistency of the fuel stock, whether the fuel is used in a compression ignition engine or a spark ignition engine, and the sophistication of the pollution control system.

Trust this gives you a lead in regard to all the issues.