When doing woods, palace, munitions factory, or citadel for example players should have the option to heat items similar to the way players can heat items at the start of every arcade depth. In order to prevent gear swapping the heat forge could be placed at the the start of depths instead of a footlocker.
Edit: I forgot about prestige missions... This would be a decent idea for them too.
No. One of the benefits of the Forge is it actually gets people out of the stupid rut of doing boss runs day in, day out.