Why would a 1996 Dodge Neon have no backup lights with a new switch on the trans fuse and bulbs?

I just solved the same problem. In my case, the replacement switch was poorly designed.

There are three wires going to the backup light/neutral safety switch. You can use a test light or volt meter to confirm you have voltage being supplied from the fuse to the switch by measuring the voltage at the connection corresponding to the white wire (ignition ON). While the connector is disconnected, you can check the rest of the system (assuming the white wire is hot)by jumping the outside connections on the connector (NOT THE CENTER!) with 12 ga. wire while an assitant (or a mirror) tells you your backup lights are on (be careful not to short the wire to ground, it will at least blow the fuse, cause sparks and other nastiness). If so, the switch is the culprit, If not, your troubles are in the wiring between the connector and the lights in the trunk.

If you pull the switch out, you'll see that the switch has a metallic pin in the very center (inside the transmission), that must be all the way OUT for the backup lights to work. Remove the switch with a 1" box wrench (you can get there from underneath, without jacking up the car even), plug the connector back on, and with the ignition ON, push the pin in and out. When the pin is out the backup lights should light up. If not, and jumping (see above) the connector worked, your switch is "no good".

If the lights light up with the pin out, but it doesn't work when installed on the transmission, it could be, as mine was, that the new switch is TOO LONG, and doesn't ever allow the center pin to come all the way out and light up the lights. I used a for-real Chrysler switch from the junkyard, and now all is well. Wish me luck returning the switch I bought.

If you are curious, you can see how it work internally by placing a mirror strategically and shining a flashlight off the mirror into the hole where the switch goes. You'll see a spot, when the tranny is in reverse, where the plastic has a dip in it to allow the pin to come all the way out. If the switch is too long, the pin never goes all the way out despite the dip, and the lights remain dark. You can also see how the neutral safety works too!

Here's an alternate solution if the switch is too long. Put a shim of some kind on the switch like, a washer, to push the switch out farther.