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Maybe none, maybe some. If you are getting a bigger "start up" gain by more tightly coupling the resonator to the active device, then the loaded Q will be lower and the phase noise will be worse.
If you are just adding more gain in the form of higher gain device and keeping the resonator loaded Q the same, then there should be no difference.
What DOES make a big difference is what happens in the steady state. In a FET oscillator, for instance, running the device heavily compressed causes trapped electrons to be released, looking like excess shot noise. An active or passive circuit that reduces the loop gain when the oscillations reach steady state will allow the device to operate more linearly, and you will have lower phase noise. This is a fairly common trick in superlow noise crystal oscillator design.
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