The Heat Metal spell description says:
Choose a manufactured metal object, [...] Any creature in physical contact with the object takes 2d8 fire damage when you cast the spell. Until the spell ends, you can use a bonus action on each of your subsequent turns to cause this damage again.
[...] If it doesn’t drop the object, it has disadvantage on attack rolls and ability checks until the start of your next turn.
If a mounted character wearing metal armor has heat metal cast on it, would their mount be considered to be in contact with the rider's Plate armor (or any other armor such as Scale Mail)?
Scale Mail. This armor consists of a coat and leggings (and perhaps a separate skirt) of leather covered with overlapping pieces of metal, much like the scales of a fish. The suit includes gauntlets.
Plate. Plate consists of shaped, interlocking metal plates to cover the entire body. A suit of plate includes gauntlets, heavy leather boots, a visored helmet, and thick layers of padding underneath the armor. Buckles and straps distribute the weight over the body.