This has been clarified in Pathfinder. You must have the proficiency to wear an armor of any type, regardless of material. Full Plate is heavy armor, a cleric cannot wear without penalty unless they take a feat. It doesn't matter if this armor is made of steel, adamantium, mithril, or cheese.
Pathfinder Pathfinder Accessories Subscriber; Pathfinder Roleplaying Game Superscriber
Yes. Even though it only encumbers as a much lighter armour, mithril plate mail still needs special training (Heavy Armour Proficiency) to be able to move fluidly in it.
I find it's often worthwhile for clerics to burn a feat to get heavy armor proficiency, since they have few traits that depend entirely on speed or mobility (such things are always useful, but not to the degree they are for a rogue or barbarian).
From the rulebook p.155:
"A character wearing mithral full plate must be proficient in wearing heavy armor to avoid adding the armor's check penalty to all…"