Let me first layout what I have. It's a 1988 1/2  Samurai that has been converted to fuel injection with a 1989 system. I drove it home when I bought it (|removethispart|@50 miles) and it was not driven again except for starting it and moving around in the yard so I could mow around it. I've spent the last 18 months or so completely restoring it from the frame up. Last week I finally finished connecting the wiring back and filling it with fluids to try to get it started.  I'm 95% confident that the wiring has been re-attached the way it came out. There were a couple of loose wires that I couldn't find a place for but they looked like they weren't connected to begin with. So I try to start it, engine turns over good but it doesn't even try to start. I poured a little gas into the throttle body and after it set for a few minutes, it fired up and ran until the fuel was burned off. I checked the fuel pump relay and the pink wire has 8.5 Volts but the relay isn't pulling in. Bought another relay with the same result. Then I noticed that as I shut the key of, when it was in the accessory position, I can hear the fuel pump running for a few seconds. When I go from off to accessory, I get nothing. But when I go to run or start and then back to accessory, the fuel pump runs. I checked voltage to the pink/black fuel pump wire when I went back to accessory and I get full voltage to the pump. In run or start position I get no voltage to the pump. I jumped the blue/black and pink/black wires on the firewall and the fuel pump runs. I have fuel up to the throttle body but it won't start nor does it squirt fuel into the throttle body. I took the fuel injector out and put voltage to the leads and it seems to be moving. I don't know how much it should move but it does move. Anybody have any suggestions? I'm leaning toward replacing the ECM but that's an expensive guess.

Don't - as in - DO NOT - "put voltage" to the injector, not unless you're planning on buying a new one soon - this particular system switches both sides of the injector and does not apply 12V to it, so there is a very real danger on damaging it should you do that.

Next - go to ack's FAQ and find the diagrams you need and verify the wiring - from what you say, there is a good chance that something is incorrectly connected - the accessory position should have no connection to the ECU, which is what controls the fuel pump, so if you're getting the ECU to turn the pump on in the accessory position, then something is wrong.

The thing is, all the engine wiring that was disconnected and then re-connected was male-female connectors. In other words, there was no cutting or splicing. The vehicle was running when I disconnected everything, so that's why the current problem is so perplexing. The only questionable connection is the idle air control. The two wires that correspond with the idle air in the connector( lt.green/black and blue/black)were pulled out. The wires that come off the idle air solenoid are red and white so I'm not sure which wires connect to which color. I've tried it both ways with no difference. I can't logically see how this would have any bearing on the fuel pump relay but nothing surprises me. The ignition works perfect in the start mode. The buzzer comes on when you first turn the key on so aside from the funky fuel pump relay situation, everything else seems to be correct. I've chased wires using the schematic I found on ack's and I can't find anything that looks wrong. If anybody knows what meter readings I should get at the injector, idle air, etc., I'd appreciate the info.

I would check inside the ECU for signs of damage, if its bad it can be rebuilt by a specialist. Also check the IAC is good, it may be that you have a short there and that was disconnected to stop it pulling the down the 12V inside the ECU.

Does any of the Dash lights come on when you go to accessories?  Check power to fuel pump relay coil when in accessory, should be "0". 
Recheck wiring.  Sounds like accessory voltage is used to pick up Fuel pump relay, but main pwr relay has no pwr to run fuel pump.  When on is selected main pwr relay picks up and pwr goes to fuel pmp, but computer has timed out and coil opens so relay doesn't allow pwr to fuel pmp.  When you go to start, no signal on accessory so no signal to fuel pmp.  Going back to on from start gives new signal to fuel pmp, and with master relay already picked up fuel pmp can run for 2 seconds.  Going back to accessory main pwr drops out.
It is close to what you described.  Just using Tracker knowledge and wiring to justify what is happening.  Go back and recheck your Main Pwr Connections.

Done some more testing. Only gets worse. Sent my ECM to Digitalguy in MN, and got it back today. While I waited for him to get it done, I removed my dash and rechecked all the wires. I took the harness out and chased/checked all the connectors against the wiring schematic, tagged them, and reattached it to the back of the dash just like it was from the factory.Put the ECM back in, double checked my connections, hooked up the battery, turned the key...same problem. But, I've found out that not only does the pump run on accessory, it runs when the ignition is off. It even runs when I unplug the ignition from the harness. So to recap:
Ignition off - no pump
Ign. accessory- no pump
Ign. run- no pump
Ign. start- no pump
Ign back to run- no pump
Ign. back to accessory- pump runs
Ign. off- pump runs
Remove fuse -pump runs
Disconnect ignition - pump runs
Unplug fuel relay - no pump

Unplug harness from ECM- pump works just like it's supposed to except it doesn't go off as long as the key is in run or start position. But as soon as the key is in accessory, pump stops.

Started checking voltage. So I put the ECM back in. Turned the key to run then off again to get the fuel pump running.  Put my meter probes into the main relay connector at the black/white wire and then to ground. As soon as the negative probe was grounded both relays opened (main and fuel). Did the same thing on the fuel pump relay. Put the +probe into the connector at the black/white wire and then touched my - probe to ground and the relays opened. The opposite side of the main relay coil is a black wire that goes straight to ground. The opposite side of the fuel pump relay coil is a pink wire that goes to the ECM. I cut the pink wire and then turned the key to run and the main relay pulled in just like it was supposed to. The fuel pump relay though didn't pull in. I grounded the pink wire that I had previously cut coming off of the fuel pump relay (kept the pink wire from the ECM hanging loose) and it too then started to work exactly like it was supposed to. So the culprit is the pink wire. When I check voltage on the pink wire to ground I get 8.5v with the ignition on. I assume there should be no voltage on the pink wire since it is the ground leg for the fuel pump relay, but I don't know what goes on after it goes into the ECM.

While I had the pink wire grounded and the fuel pump running, I tried to start the engine but the injector wasn't spraying any fuel.There are no sensors that detect fuel pressure so I thought the injector would fire regardless of fuel present. Using a 12v bulb and socket from an old turn signal, I rigged up a noid light and wired it to the red and yellow wires that connect to the injector, but I didn't get any light when I cranked the engine over. So it looks to me like I still have a ECM problem.

correction:

When the ECM is unplugged, the fuel relay doesn't pull in but the main relay pulls in and stays in until the key is turned back to accessory or off.

Never mind. Found a loose ground and that fixed everything. Go figure

There may be multiple ground wires missing "from the conversion" - the only way you would/should see symptoms of that nature resulting from a loose ground wire is if it is the only ground wire, and the should not be the case.

I'm not sufficiently familiar with the EFI Samurai to tell you how many grounds there should be, but the following are common on most vehicles - from the battery negative, two cables, one to the bell housing, the other to the body, the EFI system will typically have a ground to the body and also a second one usually to the intake manifold, there may also be a completely separate ground between the body & the manifold.