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.
Best Answer
Proficiency Bonus
When you wear armor, you apply your proficiency bonus based on the armor you are wearing (pg. 274, Core Rulebook, "Armor Class"). In your example, the Barbarian may equip heavy armor, but won't be able to add their proficiency bonus to their AC.
At level 1 your proficiency bonus may only be a few points different from being untrained. However, over time this gap widens because your level is a part of your proficiency bonus and because of differences in your degree of proficiency. This certainly doesn't keep you from equipping heavy armor, but it will quickly become comparatively unattractive.
In your specific example, it may be important to note that some Barbarian feats have armor restrictions. For example, Predator's Pounce (pg.92, ibid.) requires that the user is unarmored or wearing light armor. Proficient or not, you couldn't use this feat while wearing heavy armor.
Armor Check Penalties
In Pathfinder 1e there were penalties to wearing armor you weren't proficient in. As summarized in the answer to this question, this includes a penalty on attack rolls and some skill checks. This doesn't exist in Pathfinder 2e. Instead, it's covered by the proficiency bonus. There is a check penalty in 2e, but it applies to characters with insufficient strength to effectively use their armor, not their proficiency (pg. 274, ibid, "Check Penalty").