No need for total oxygen evacuation. You actually just have to drop oxygen content in the air down from 21% to 15% to make most substances (including oil and grease) inflammable, and this air is perfectly safe to breathe for humans.

In fact, if you consider the partial oxygen pressure (which is what matters for your lungs), this air is no worse than the air you find inside an airplane, or at a 2500m (8200 ft) mountain peak; Wikipedia says 140 million people live permanently at above 2500 m.

Now, this already exists and is called a hypoxic air fire prevention system, and is used e.g. in some data centers and in archives of valuable documents. It's not used in fast food restaurants though, because they have better ways of preventing and extinguishing fires at lower cost.