One consideration on running a 240 volt transformer from 120 is that the windings have the same number of turns and the same wire size so the windings have the same resistance values on either voltage. If run on rated 240 volts, the 24 volt secondary might have 2 ohms of resistance (probably less). If you had a 1 amp load then the 2 ohms winding resistance would drop 2 volts and the output would be 22 volts under load or about 92% of the no load voltage. Transformers may be rated for full load current but the 'sag' will be proportionally greater when run at 1/2 voltage.
If you run the same transformer on ½ the rated voltage of 120, giving 12 volts out, with a 1 amp load, the same 2 ohms will still drop 2 volts and the output will be 10 volts or 83% of the ‘rated’ voltage. The same effect happens on the primary where the greater resistance of the primary winding will have an effect on the voltage the primary sees but where the load voltage will be ½, the effect will be less. So the voltage out under load with 120 volts in might be 75-80% of the no-load voltage as opposed to around 90% with 240 volts in and the no-load to load regulation will be worse.
Generally these losses won’t be a problem and I have used transformers at ½ rated voltage and they run nice and cool.