Hey guys! Long time Ram owner here. I've owned my 2005 Dodge Ram 1500 Daytona Quad Cab SLT since new. #188.
I replaced my ignition switch and had the same issue. Ignition chime will not stop regardless of battery being disconnected and resetting any and all fault code condition. These situations always seem to stem from what was done or what changed. I just put up with it for several months now. Well today I decided enough is enough... pulled the ignition switch back out to check the connections (I figured something isn't making contact that is supposed to). My original ignition switch had the upper right terminal burnt (Looking at the harness connector straight on from the driver's seat) so I decided to bend the Copper spade connections slightly toward the connections in the harness i.e. to the right (passenger side) as they weren't real straight as received. I wanted to make sure I had intimate contact with the harness connections.
Also, there are two small plugs just below the bracket that holds the steering wheel tilt arm/lug. One is a yellow plug with green wires, and the other one is a white connector with various wire colors. The one to the right (yellow plug) I know from experience is for the steering wheel stereo controls, the white one that has to be disconnected in order to get to the ignition switch screws I'm not totally certain what it is responsible for doing but there are 6 various color wires going to it (white, black purple wires etc.). It may have the ignition chime circuit in it. I made sure it was plugged in properly as well as making sure the wires feeding into the plug were seated properly (wiggled the wires, pushed them in etc.). Viola! NO MORE annoying chime with the key removed and driver door open!
I hope this helps some of you.