I'm having the same exact issue. I'm curious to know how you Diagnosed the unit to have a bad cat?

wow a plugged up cat... who woulda thought of that one?
lol
glad you got it figured out :o)

I really appreciate everyone's help. The forklift is now up and running. It ended up being the catalytic converter being plugged up creating too much back pressure to start.

reading all responses and comments i have to ask. Have you checked the pushrods? may have one bent or a valve not opening. To me is sounds mechanical. if you remove cylinder 1 plug, you are basically removing the compression. And you say fire. are you losing spark or just wont start?

i haven't seen you mention anything about hooking up the PSI Display program to the truck (or whatever toyota uses, i'm sure it is the spectrum program though) to see what is actually going on in the ignition system.
Are you absolutely sure the plug wires are connected in the right firing order?
This is the strangest problem i've ever heard of, EVER (and i've seen some strange ones over the years).
You did get a change though by replacing the whole distributor/ignition module so there's that...
You may have to go back to the start and verify fuel, air too and make sure there's enough fuel getting into the engine and there are no intake problems, valve adjustment issues.
Just start from the beginning as if you were just setting up the engine to start it up for the first time, set the timing to #1 TDC, check distributor position, rotor position, plug wire connections, etc etc.
This is strange indeed...

We just replaced the distributor, ignition module, & plugs with a brand new. The engine will start with all plugs in but will immediately die, sends a burst of air out the intake tube. Will run for about a minute if we take out cylinder #1 plug.

It seems like nothing makes sense i would put a new or known good distributor in it who knows might cure problem

We have gotten to a point where it will run with cyl 1 plug removed. It will start with the plug in cyl 1 but will immediately die.

This is a problem l see you called the dealer and they could not find the problem. Could you try disconnecting the ignition module plug to the est there is two plugs one with two wires and one with four remove that plug it will still get spark but will not start because Toyota controls the lock off from this plug so drain down the gas and just spray a small amount of starting fluid and see if it starts up for a second that should tell you if the ecm is giving the dist the wrong timeing.if it fires up with the plug with the two wire l would say you have a ecm problem get back to me.

Tell me will engine turn over with #1 plug in ?

We installed a new Delco ignition module, coils and plugs. As well as adding an extra ground. The cap is a D328A. We are still having the same issue. We also had a technician from the Toyota dealer come take a look at this forklift, he was not able to track down the issue unfortunately.

tell me what did you find to fix or still looking

Are you using Delco ignition parts or TVH?
Removing a wire forces coil to full output. Coil is bad, module too. Use Gm Delco parts
Flat cao or tall?
Flat cap" D328A delco
rotor ' D465

Tall cap D313A Delco
rotor' D446

Swoop is right one plug would not stop this truck from running l would ask you if you leave the spark plug in but disconnect the plug wire so no spark is going to that plug will the truck run. If so then l would say you have so kind of spark problem.l myself had a problem not what you are having but a problem on the same model truck approx a year ago the truck would only get spark if l disconnected the est side of the ignition module l was thinking it was a bad module but it turned out to be a ground problem Toyota had ground problem on some of these trucks you could try hooking a extra ground from the neg of the battery to the engine block and see if it starts

well then , this is a true mystery from the sounds of it.
there wouldn't be any plausible reason for the engine to not run just because of the #1 plug being in place. I would go back into the ignition/distributor, double check timing etc. Something is causing that, something you haven't stumbled across yet.
Maybe it's something in the ignition module in the distributor.