You are asking a reasonable question that doesn't have a reasonable answer.
Potentiometers, rheostats have quite a variety of uses in circuits, which as was stated before can't be boiled down to a general rule.
Let's take a simple control circuit, in which potentiometers are commonly employed for manual variable adjustment. The variable to be controlled could be volume, speed, temperature, direction, frequency, degree of movement, etc. As many variabilities as you can imagine can be listed.
So, lets take your vibrating bowl as an example. You say a 220 ohm rheostat controls how violently it vibrates upon a spring. It used a magnet, but I'm imagining it also uses a coil or transformer to electromagnetically oscillate the magnet within the coil's field, the spring being there to dampen the movement. If it is very simple, it is using mains AC, in the US this would be 120 volts @ 60 Hz.
If the rheostat is controlling the current through the coil, which it likely might be, the amplitude of the oscillation is what is being controlled. A higher current would produce a stronger magnetic field around the coil and more forefully move the magnet. A slighter current would oscillate the magnet less forefully because of the smaller field.
OK, so in a typical circuit the 220 ohm value would be maximum resistance and might set minimum amplitude in the oscillation. The rheostat would go down in value from there, and inversely the amplitude of the oscillation would go up until it is maximum @ close to zero ohms.
Now, if you replaced this with a 100 ohm rheostat, your minimum oscillation wouldn't be as slight. If you replaced it with a 1kOhm rheostat there would be a significant degree of pot shaft rotation that would be effectively "dead" because the resistance becomes too great and the magnetic field is too slight to overcome the spring. In that case, from 220 to 1000 ohms you have little to no effect on the bowl's vibration.
This is an example where the consequence of changing the rheostat isn't dangerous, but it is somewhat foolish because you are potentially reducing the performance of the appliance.
And note, my "description" of the circuit is totally assumed and not implied to be the actual method of operation for your vibrating bowls. Don't go off and say its ok to change anything!
In other applications, and quite possibly this one, the consequence of changing the pot could be damaging if not dangerous.