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.