1999 dodge neon.. My fans will not turn off

This all depends on what wires are broken. The most likely reason for the fuses popping is because of something shorting out against something else. Whether that be wire to wire or wire to frame.

Here's a link to the P0340 code,and P1491.Powertrain Diagnostics Manual P-1491 Radiator Fan control relay circuit
Set condition: An open or shorted condition is detected in the radiator fan relay control circuit.

Possibilities:
Fan relay ground circuit open
Fused B+ circuit open
Rad fan control relay defective
RF relay output circuit volt cycles over to batt voltage
RF relay control circuit open
RF relay ctrl rly ckt wir harness observable or intermittant defect
RF relay ctrl ckt short to ground
Defective PCM

Get out your Multimeter.

The dodge neon fan switch is located on the front left inner frame behind the radiator, Take note this relay doesnt look like a normal relay, it looks like a module type, heres a picture of it. also take a look at the connector and make sure its not melting or rotted. the most common reason for a relay like this going bad is due to a bad fan, it draws to much, make sure the fan is good, replacing the relay with a bad fan will make the relay go bad in a few months,good day, hope this helps.

YOU NEED TO CHECK DARK GREEN/ORANGE WIRE AT IGNITION COIL WHEN CRANKING ENGINE FOR POWER..IF NO POWER,SHUTDOWN RELAY IS NOT BEING ENERGIZED..THIS ENGINE ALSO HAS CAMSHAFT POSITION SENSOR ON END OF CYLINDER HEAD..ENGINE CONTROLLER NEEDS CAM AND CRANK SIGNALS TO ENERGIZE THE SHUTDOWN RELAY..UNPLUG CRANK SENSOR AND CHECK VOLTAGE ON ORANGE WIRE..SHOULD HAVE APPROX 8V..IF NOT..UNPLUG CAM SENSOR AND SEE IF 8V SUPPLY RETURNS..IF NO POWER TRY UNPLUGGING VEHICLE SPEED SENSOR LOCATED IN TRANS HOUSING WHERE RIGHT SIDE AXLE COMES OUT OF TRANS..IF 8V RETURNS..THE SPEED SENSOR IS SHORTED INTERNALLY..THESE 3 SENSORS SHARE THE SAME 8V SUPPLY CIRCUIT..WITH SPEED SENSOR UNPLUGGED,PLUG CRANK SENSOR BACK IN AND TRY TO START ENGINE..HAVE SEEN THIS SAME PROBLEM AT WORK RECENTLY

the wiers for the fule pump come up through the floor under the rear seat.

The timing may have jumped,Have the timing checked.

It still could be numerous other problems,so post back with details

Well it could be the pump itself going bad or a short in the wiring, possible clogged fuel filter. There is a valve what looks like a tier valve cap on the fuel rail if you can get you hands on a pressure gauge and hook it up, you should see the pressure increase and decrease, if it does not start looking elsewhere.

could be signs that you need to replace the radiator. neons have a rpoblem with the radiator cores rotting out pretty quickly.

According to some Dodge Neon forums, theres a wire in a harness that gets rubbed through and shorts out. It is located on the back side of the engine near the cylinder head.

They also say it could be the fuel pump relay that is sticking.