Some crystal oscillators have an enable pin. On some of those the oscillator turns off completely, some just shut down the output stage.
If the oscillator shuts down, you need to allow up to 5 ms for it to start, but you need to do that with any crystal oscillator that has stopped.
You can supply a crystal oscillator from the output of a PIC, as they only use about 5 mA, and then just shut it down when you don't need it.
If you don't need a high clock speed, you can get a 32.768 kHz watch crystal to oscillate using just a few :mu:W. If you do shut those down, be aware that they can take several seconds to restart.
Resonators are used like crystals, but they usually have the capacitors inside so that you don't need to use external ones. They are not as accurate as cystals but they start a lot faster.
If you use a PIC and a crystal, the oscillator shuts down in sleep mode.