I have a cheap, 40W 120V soldering iron with no temperature adjustment. Given the internal circuitry seems pretty simple would it generally be safe to run it on 240V?
A big NO. What would you expect a 120V lamp to do if you powered it from 240V! The results could be similar.
A simple soldering iron is just a resistor - if you put twice the voltage into it it will produce 4 times the heat - rather than heating to 700°F it could get to 1000° or more. It could cause a fire, the insulation could break down and cause electrocution danger.
You could run it from a step down transformer - one capable of powering it should no be expensive but neither would a 240v soldering iron.