Charlie was (to use a current term) my mentor in the LP business most of what I know about LP is because he taught me.  Charlie said "Whenever a propane fueled engine starts acting up to rebuild the vaporizer.  Even if that isn't the problem it will help."

Most LP motor fuel systems work off a vacuum. The engine has to pull enough vacuum to actually draw the propane into the carb.

Have you tried priming the carb?  On the vaporizer there should be a button right in the middle like this IMPCO

It may have a solenoid over it like this Century 

  If no solenoid try this.  Turn on the gas tank, push in the primer button until it gets hard to push, then try starting the engine. 

If it has a solenoid you'll have to find which switch activates it.You should be able to take 2 - 4 bolts out and remove it to see if it is working BUT IF YOU DO NOT KNOW LP SYSTEMS LET SOMEONE TRAINED OR EXPERIENCED DO IT.  It isn't worth risking injury or explosion.  It isn't rocket science but it is a hazardous material and can quickly injure you if you make a mistake.

(photos borrowed from http://www3.shopping.com/xDN-building_supplies-propane_adapters)

and instructions.

FIRST with the engine running turn off the LP tank.  It will run forever it seems. Once it dies turn off the ignition switch and go do something else until the coolant is safe to work with. I suggest making up some soap solution.  Your LP guy will have the good stuff.  But if you decide to make your own I suggest using shampoo instead of dish soap.  Dish soap often has ammonia in it which messes with brass fittings.

You will also want disposable gloves.  Because the oily stuff you are going to find inside will stay with you for days.

The next step will be put on some LP/anhydrous gloves.

If you don't have any dipped, lined, water and chemical resistant lined gloves you can improvise with a pair of well insulated gloves, maybe with unlined rubber gloves over them. Eye protection is a good idea as well.  You just want to be careful disconnecting the LP lines in case there should be pressure in them.  Liquid LP gets to -44 VERY quickly and does really nasty things to exposed skin.

There may be a fuel lock off which should have a filter inside

This one does not have a filter

You ought to replace the filter while you have the system down.  

I'm not trying to be a lawyer with all the side-stepping, cautions, "may's" and "should's".  It's just that A) there's a lot of variations in equipment B) who knows what modifications someone may have performed on it and C) there are undoubtably folks reading this who really ought to let someone else do the work.

We used mostly Impco  vaporizers with VFF30 lock offs.  And while we worked with LP every day with no problem, an untrained or inexperienced person without a "Charlie" standing by could do something D&D (Dumb & Dangerous)

A site I stumbled across while writing this is http://www.propanecarbs.com  Use your own discretion, I know nothing about them but the site looks helpful.

The actual work is a no-brainer if you follow the instructions. I really recommend the disposable gloves ... and throw away coveralls. And do it Monday morning so the smell wears off your skin a bit by Saturday night.

I had a route customer with an old Ford pickup on LP that would start at -20 setting outside with no heat. He was an exception.  Charlie taught me (back to Charlie again) if it got much below freezing to plug it in and always let the engine warm up a little before trying to drive it.

I just realized my first experience with LP as a motor fuel. My best friend in High School's dad managed a local LP plant.  I remember pulling in his driveway one late fall day to see Bud pouring hot water on something under his truck's hood. He said he knew he should have let the #$%^ thing warm up first.  Bud lived across the street half a block from the office.  He was only crossing the highway so he didn't let it warm up first.

3 basic facts you need to know to understand the problem

1)Liquid propane does not burn. VAPORIZED propane burns. For reasons we won't dive into here propane is taken from the fuel tank in liquid state.

2) When a liquid held under pressure above its boiling point is released to atmospheric pressure it boils and it makes the temperature its boiling point.  If you use a pressure cooker to heat water, when it escapes as steam it is 212F.  The boiling point of propane is -44F.  So when it changes from liquid to vapor it is -44.

3) Firefighting 101 teaches us fire requires heat, oxygen, and fuel. 

How is all that relevant? LP engines run on vapor and water cooled engines generally use a coolant warmed vaporizer to get the propane warm enough it is a vapor state.  When it is cold the propane does not vaporize well.  Until the water in the vaporizer is warm your engine's fuel supply is lousy.  It may have droplets of liquid that do not burn well and when they turn to vapor rob heat out of the combustion chamber.   This liquid propane is robbing the combustion process of 2 of the 3 requirements for a fire.  No fire, engine no run.

Rebuilding a vaporizer may help that.  As the vaporizer is used it builds up "crud" inside.  The crud keeps it from working like it should.  Back to "Charlie said" ..cleaning the crud out is the most important part of a vaporizer rebuild.  So if your forklift is not running well until it is warm you might try rebuilding or replacing the vaporizer.

I can't address how it was different because I never got into that aspect of it. If it got to that point I was out of Band-Aids and we sent the patient off to a mechanic.  I remember being told this is why when converting a computerized engine you have to put in a box that converts what is really happening to what the computer wants to see.