What happens when you wear armor you’re not proficient in?

What penalties or restrictions does a creature have when they don armor they’re non proficient in?

For example, picture a newly adventurous (level 1) Dwarf Barbarian (18 Strength, 10 Dexterity) who has inherited a set of Full Plate armor. He’s only Trained up to Medium Armor, but he feels obligated to don it. What are the results? What if he still wears this armor at level 10 (assuming he never takes the Armor Proficiency Feat)?

This is generally a question to clarify related rules, but it has the possibility to come up. For instance, said Barbarian may invest in Feats that give benefits when hit or critically hit. Additionally, you could ‘equip’ the enemy Wizard you’ve captured to add to your restraints.