I know there was a shortage of Oak to some extent in England and Wales during the expansion of the Navy in the 18th century but I wondered if France, for example, ever looked to destroy any forests to further hamper this.
Not that I have heard of. It was common practice to attempt to interdict supplies, especially "naval stores" (products derived from pine sap) from the Baltic or the Americas by capturing merchant ships, but I've never heard of an attempt to deforest areas and I'm not sure how you would accomplish that with 1700s-1800s technology.
In any case, demand for wood was one reason that European countries built ships in their colonies. Here's a longer post I did about that awhile back.