If there is a product that can damage wood furniture, I've probably done it. Even water can ruin cherry wood's finish. I would worry much more if BP or any fabric spray damaged Formica or tile or a non-wood counter top.
And yes, I love BP with the lightest scent only. But I did have a hand-dyed red fabric bleed across the seam line of a scrappy quilt when I failed to press it soon after spraying.