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 will have lower drive levels but they don't get damaged quickly by overdriving. They can age faster if overdriven.
As for what the circuit is doing, it looks like an unmodulated transmitter. There is no control of the amplitude and the direct connection of the antenna to the oscillator circuit will make the frequency unstable. Two of those in the same room will phase-lock to each other. There will be a lot of harmonics generated.
If that is being used as a transmitter, there is unlikely to be any legal way to use it outside of a screened room.