Can I wire a dimmer switch into a soldering iron?

The problem is likely the 3-wayness of your dimmer switch. It sounds like you are connecting power and the iron to the runners travelers, instead of through the output pole terminal.

           ___[traveler]____
hot _____./                 \.____  switched wire
           ___[traveler]____      |
                                  |
                                  O [light socket (on)]
                                  |
                               neutral

           ___[traveler]____
hot _____.                  \.____  switched wire
          \___[traveler]____      |
                                  |
                                  O [light socket (off)]
                                  |
                               neutral

Make sure you don't have your's set up like this:

               ___ pwr
wirecap _____./       
               ___ iron

Also make sure you have clicked over the dimmer to the correct traveler or wire the two travelers together (the switch will always be on then).

of the There are some electronic dimmers that detect the other switch's position and go to full brightness, which would probably not work correctly if only partly wired.

You could look at a simpler dimmer switch or cord like this http://goo.gl/Mnwbe, though as other's mentioned in the comments, it's probably only $20-30 more to get a new variable wattage iron.

Whatever you do, please be careful.

Yes, you can. I did it with a iron solder sucker. It does reduce the voltage to the iron, however that alone does not control the temperature. The temperature will continue to rise to some maximum. That value will be a lower temperature than with a higher voltage. So yes it does work, only not like a temperature controlled iron.

I mounted the dimmer in a project box, and added an outlet receptacle in the rear to accommodate the iron.

Ofcourse you can. Dimmering the solder iron won't alloy you to set temperature directly, but with some little practice you'll be able to obtain great results with a cheap iron. Pofessional soldering sets allow you to set temperature in degrees but who says which temperature to use ? The same practice that will tell you how to adjust not degrees but dimmer scale. Another thing the professional set can do is to control temperature by a sensor and feedback loop. This is a nice feature that a dimmered iron won't have but again, with some practice you'll be able to overtake this.

Note the dimmer's point where your iron still can melt the soldering alloy. Keep the soldering iron on this position or a little lower when idle, or to solder smallest pieces, increase it according your experience as you need more power to solder bigger pieces. If the alloy can't stay on your iron's top, you're using too much power.