Can I use a router speed control to vary the temperature of a soldering iron?

Certainly if the iron plugs directly into 120volts.

Perhaps but why not buy a temperature controlled Weller soldering iron? It comes with a little sponge to wet and clean the soldering iron. Weller calls it a soldering station and has several soldering tips easy to change.

If it's a cheapo Radio Shack iron, yes. More advanced irons costing more than $100, then probably no. The expensive irons have temperature regulator controls that actually control the tip temperature.

Maybe.  But even if it does work, you are not regulating the temperature, you are only regulating the dissipated power.