Although George Washington did wear false teeth, they were not made of wood. He had multiple sets of dentures that were made out of materials like gold, ivory and lead.
According to the Mount Vernon Library, the origins of the myth of the wooden dentures is unclear. However, it is likely that it comes in part from the way his ivory dentures became stained and worn over time, which gave them a grainy, brownish appearance that resembled wood.
By the time of his First Inaugural Address, Washington had lost all but one of his natural teeth. Dentures at the time were clumsy and uncomfortable, so he was often bothered by mouth problems during his presidency.