Why isn't fuel pump pumping?

On the pump harness, the pink wire goes to the sending unit and the gray or tan/white goes to the pump.
On the relay, the green/white grounds the relay from the engine computer, the orange is hot, the red is a jumper wire, and the gray goes to the pump. Keep in mind the relay is only "on" for 3 seconds when you turn the key on. The engine computer must see RPM or oil pressure to continue to power the pump relay.

Fuse? Relay? This is suppose to turn a relay on to pump fuel when key switch is turned on. Then when engine is started as long as there is oil pressure it will keep running.

Well, the red wire is what you use to put power to the pump when the relay is off. Not when the relay is on.
When you turn the key switch on, you are putting power to the magnet of the relay on the dark green and white coming from the computer. The magnet closes the switch inside the relay. The orange wire powers the pump and the gray wire goes to the pump. When the switch closes inside the relay, you complete the connection between the orange and gray wires.

I don't usually do diagrams due to copyright laws.
From the fuel pump you have a gray wire and a black wire.
The black is ground and the gray is power from both the pump relay and the oil pressure switch.
The oil pressure switch provides power to the pump once the engine starts.
At the relay, you have a green and white wire that provides power from the ECM with the key switch on. You have an orange wire that provides power from a 15amp fuse.
The gray wire going to the pump, and a black and white wire for ground for the relay.
The ECM provides power for 3 seconds when the key switch is turned on to boost line pressure for startup.

You may be geting gas from the fuel pump but are the injector working or do they have power going to them. You can check the dark green & orange wire at any injector. Also check for power at the plug of the coil (there are two together but both get power from one place). The power wire for the coil also happens to be dark green and orange. If you are not getttting power to either one, check the auto shutdown relay (ASD). This relay stops power to the injectors and the coil but the fuel pump will still work. It doesn't shut down the fuel pump. You can let me know here or you can email me directly at [email protected] with all your information and I will be happy to help and no there is no cost. Just need to be sure if this is the case first (the ASD relay) to go any further. Also if you have a turbo this information isn't correct as far as the wire coloring goes.

Are you getting battery power and ground at the fuel pump connector? (This should have been checked BEFORE replacing the pump)

Other things to check:
1. Fuel Pump relay in the Engine Compartment Fuse Block
2. The relay is powered (Secondary -or- "fuel pump side") by Fuse 0 in the Engine Compartment Fuse Block.
3. The relay Primary -or- "control" side is powered by Pin 37 RED wire at the PCM
4. The relay Primary control (or ground circuit) is operated by Pin 22 LIGHT BLUE/ORANGE wire at the PCM
5. The Inertia Fuel Shut-off Switch located between the fuel pump relay and the fuel pump may need to be reset. The switch is located behind the right kick panel. The DARK GREEN/YELLOW wire at the switch is the feed from the fuel pump relay. The other wire goes to the fuel pump. Some units have the DARK GREEN/YELLOW wire going in and the wire going out can also be DARK GREEN/YELLOW, or PINK, or ORANGE - the wiring diagram for your vehicle does not specify the color of the wire that actually goes to the fuel pump itself.)
6. If none of the above produce any results, make sure that your ground wire at the fuel pump has good continuity to ground. It usually screws to the left frame rail near the fuel tank.

Any of these components (including the PCM) or their connecting wires could be the cause of your problem.

You didn't post an engine size but you are correct in thinking there is a fusible link that blew when you grounded the orange wire but there is also an inline fuse that protects the oil pressure switch and the fuel pump relay. It is usually located on the firewall.

You can check the fusible links near the starter for continuity. If one of them is burnt up start there by replacing it. If none of them are bad you will need to find the inline fuse. At least the 3.3 liter engine is set up this way.

I’m happy to assist further over the phone at https://www.6ya.com/expert/jeremy_d728a59f986299fa

Hello

Power for the fuel pump is supplied by fuse #16 in the Power Distribution Center (PDC), goes through the Fuel Pump Relay and then to the Fuel Pump and the ground completes the circuit. Power for the Fuel Pump Relay coil is supplied by fuse #18 in the PDC and the Powertrain Control Module (PCM) supplies ground. It is a fairly simple circuit to check.
The first thing to do is check all the fuses, especially #16 and #18. If you have they are good then you will need to pull the fuel pump relay and check for power, with the key ON at the dark green/black wire and orange wire. The DG/BK wire is the power for the fuel pump and the OR wire is the power for the relay coil. If you have power there connect a jumper between the DG/BK wire and the dark green/white wire. You should hear the fuel pump running. If not there is a problem in the DG/WT wire from the relay to the pump.
Okay, lets assume that checks out fine, the next thing to do is replace the fuel pump relay and ground the dark blue wire. You should hear the fuel pump running. Okay? Good, now go to the PCM located on the firewall in the engine room. It has three large connectors, black, white and grey. There is a DB wire in cavity 19of the grey connector. Ground this wire and you should hear the fuel pump running. If it does then the wiring for the fuel pump circuit is all good and the PCM will need to be replaced. If it doesn't run, then you have a problem in the DB wire between the relay and the PCM.

the other wire that is "dead" is the ground wire for the fuel pump. try cutting that wire after the connector towards the pump, and running it to a good clean ground.