I think the right technical answer is.....it depends!
A dryer outlet is 240V, and typically fused at 30 Amps. That 240V is actually both phases, or (2) 120 volt circuits, which since they are both phases gives you 240 across the two power legs. (power leg A - - - NEUTRAL - - - power leg B)
If the converter uses one of those 120 volt circuits to ground, it would give exactly the 120 VOLTS needed. However the one problem might be it's amperage capacity. Since it is most likely fused at 30 amps, it will not trip unless more than 30 amps is drawn.
A typical 120 volt circuit is fused at either 15 or 20amp, and therefore the cords to those devices would be rated most likely at 20 amps. Since you are plugging that device into a 30 amp fused dryer circuit, the situation could exist where the device has a short and wants to draw 25 amps (anything more than the 20 amps the cord can handle, but less than the 30 amp which would trip the breaker. If it did this while plugged into a normal 120V circuit, the fuse would trip and prevent a fire. While plugged into 30Amp outlet, the cord could overheat and burn up.
Now if this converter plug you are mentioning has a built in fuse that trips at 20Amps (as opposed to the 30Amps the dryer circuit is most likely fused at) then I would not be concerned. If it has no internal fuse, then there is 'potentially' a fire risk, if the device has a failure.