Can alkyl-substituted phenols be reduced to alkyl-substituted benzenes by zinc dust?
This is the basic principle of functional groups. Organic chemists quickly noticed historically, that some compounds reacted in the same way when presented with the same chemicals; thus allowing them to deduce that it was only a certain functional group (‘radical’ in historic terminology) that was reacting while the remainder of the molecule remained untouched.

Unless conditions or reagents are known to also attack other functional groups, they will typically be unharmed in a reaction and whichever features your molecule has, it will still react in the desired manner. For example, if phenol gets reduced to benzene by distillation with zinc dust, we can be pretty sure that cresols (e.g. paracresol, 4-methylphenol) will react in the same way, only that the final product will be toluene rather than benzene.

As I am not aware of zinc dust reacting with a double bond — the only other functional group present in chavicol — I am willing to say that it will indeed lose the hydroxy group and be reduced to allylbenzene as you desire.