Does running a dryer on 120 V drive up electricity costs?
Not an immediate danger, but quite a few problems
According to the spec sheet, this dryer is rated for 240V, 30A. Running it on &quot;120V outlet&quot; in the US normally means either 15A or 20A. Thanks to Ohm's law, if you run it on a 20A circuit you are probably safe. If you run it on a 15A circuit, there may be a safety issue. But in any case, there are plenty of other problems.
A dryer has three main power consuming sections - controls, drum motor and resistance heating elements. Typically (but can't tell for sure on any particular dryer without checking the schematics or the actual wiring), the controls &amp; drum motor require 120V (Hot-Neutral) and the resistance heating requires 240V (Hot-Hot). If you manage to get the dryer working on 120V, then presumably you have everything working on Hot-Neutral.
As numerous others have pointed out, thanks to Ohm's law, if you cut the voltage in 1/2 and keep the resistance the same, the current will also be cut in 1/2 and the power produced by 1/4 (power = current x voltage). The result is nominally 4x the drying time. But due to other related factors, my gut feeling is that the drying time may actually increase by MORE than 4x. But we'll go with 4x.
Assume for the moment that the dryer uses, normally, 22A @ 240V for heating. That is now cut to 11A @ 120V and instead of 5,280 watts it is only 1,320 watts.
However, the drum motor (controls are minimal power on a modern machine), will still use the same power it used before. If it was previously 5A @ 120V, now it is still 5A @ 120V, so the total usage will now be 11A + 5A = 16A. If my numbers are correct (they are an arbitrary guess and may not represent reality, but they are one possible combination) then the usage of 16A would be greater than the normal capacity of a 15A 120V circuit. So there is a possibility of over-current if this is a 15A circuit and the numbers are &quot;just right&quot;. It is also possible that this is a 20A circuit (no problem) or the numbers are a little different (e.g., heater 22A =&gt; 11A + electronics 1A + drum motor 2A = 14A total) and no problem. So there is a concern but it is NOT my original &quot;BIG PROBLEM&quot;.
As far as electricity cost: In theory, if 120V == 1/4 the heat produced and the clothes dry in exactly 4x the normal time, then your electricity costs would be the same as running at 240V. However, it is quite likely that the clothes take substantially MORE than four times as long to dry for &quot;physics reasons&quot; (I can't get into it all right now, though I still stand by that statement despite my retraction of the imminent safety issue).
Bottom line: Your current setup:

May be a real hazard due to possible over-current of wiring and equipment
Is a waste of energy (as you already suspected)
Takes way too long to dry your clothes (as you already know)
Is almost certainly shortening the life of the dryer due to use beyond the design

Get it fixed - put in a proper 4-wire NEMA 14-30 outlet, a 30A 2-pole breaker and appropriate wiring (minimum 10 gauge copper).