Short answer, no it won't as crystals are really difficult to overdrive.
Long answer, no it won't and here is why. That circuit will run that crystal at around 50 pF load capacitance so it's equivalent circuit is an inductor that tunes at 8 MHz to the 50 pF, so the impedance is around 400 Ohms.
I would expect an amplitude of around 3 V pk - pk on the crystal and so that is about 1 V rms, so the current is about 2.5 mA rms.
The ESR of the crystal will be something like 50 Ohms, so with 2.5 mA flowing, there will be around 300 μW dissipated in the crystal, which is quite a bit less than the 1 mW which is the typical maximum for an HC49 crystal.
https://portal.iqdfrequencyproducts.com/products/details/hc49.pdf
Small crystals...