One problem is that old non-grounded receptacles aren't polarized.
Polarized plugs are needed to control how the appliance is energized. Appliances that have an on / off switch must have a polarized plug.
Along with an on/off switch that's connected to the hot wire, so that the electronic devices are not energized while the switch is off. If the plug is reversible, the appliance will still function properly, but the components will be energized, even when the switch is off.
A foreign object or insect could cause a short circuit, even with the switch off, if the polarity is reversed.
It's very common to see older non-grounded outlets wired incorrectly. What will happen is that a polarized plug won't fit, or it will have to be forced into the receptacle, possibly damaging it. The polarization could be wrong. So you'd have to remove the wall plate and check to see that when you insert the plug the wide blade is neutral (white), and the narrow blade is hot (black). Even if you use a cheater plug, the polarization could be wrong.
As for other problems: I've seen electronic exercise bikes and stair stepper machines control panels not work because of the lack of a ground.
So you'll have to try each piece of equipment and see how it goes.
In the long run, I'd suggest having the entire service upgraded to a properly grounded system.
If a toaster is plugged in backwards it could possibly un-toast the bread, turning it into dough.