Let's just look at one example, dryer heater plated at 3600W @ 240V or 15A. (240V x 15A = 3600W)
Back-substituting into R=E/I, R = 240/15 = 16 Ohms.
I = E/R. If E now = 120V and R is still 16, I = 120/16 = 7.5A.
7.5A x 120V = 900W, insufficient.
However, if we consider a heatbox using all 3 240V wires center tapped to neutral, leave the neutral where it is (center) and jumper both the 240V hots to 120V hot (elements in parallel):
I = 120/8 = 15A, x120V = 1800W (heater only).
The motor on my 120V portable is plated at 4A. 120V X 4A = 480W. 1800W + 480W = 2280W.
2280W/120V = 19A, the limit for a 20A standard branch.
Whereas, my portable (designed for 120V), heater is 1450W + motor 480W = 16A, a much more suitable load for a 20A branch and I believe the max a single 120V plugin appliance is allowed to draw.
So if the dryer has a 3-wire heatbox and you jumper the hots to parallel, you cannot have ANYthing else plugged into that branch. Worse, if it sets fire to your house and your insurance adjuster discovers the modification, he can nullify your coverage.
Now then, few heatboxes anymore have 3 wires. They're either designed for one standard or the other. They were 'compatible' in the 50s, where either connection would not overload. But that would have to compromise both connections.
As above, the thing to do is buy the heatbox for the voltage you are using.