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

Question

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.

Answer 1

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").

Answer 2

Simply put, you do not gain your proficiency bonus to your AC. At level 1 this may not make much of a difference but your AC will be much lower as you level up. below is a table of how your AC will scale with light armor(studded leather AC +2, dex cap 3) and medium (hide AC +3, dex cap 2) with proficiency and heavy(full plate AC +6, dex cap 0) without (assuming 14 dex )

+-------+-----------+-------+- ------+-------+
| Level | Unarmored | Light | Medium | Heavy |
+-------+-----------+-------+- ------+-------+
|   1   |    15     |  17   |   18   |  16   |
|   2   |    16     |  18   |   19   |  16   |
|   3   |    17     |  19   |   20   |  16   |
|   4   |    18     |  20   |   21   |  16   |
|   5   |    20     |  23   |   23   |  17   | +1 potency rune, +Dex increased to 16
|   6   |    21     |  24   |   24   |  17   |
|   7   |    22     |  25   |   25   |  17   |
|   8   |    23     |  26   |   26   |  17   |
|   9   |    24     |  27   |   27   |  17   |
|  10   |    26     |  28   |   28   |  17   | +dex increased to 18
|  11   |    28     |  30   |   30   |  18   | +2 potency rune
|  12   |    29     |  31   |   31   |  18   |
|  13   |    32     |  34   |   34   |  18   | proficiency increased to expert for unarmored/light/med
|  14   |    33     |  35   |   35   |  18   |
|  15   |    34     |  36   |   36   |  18   | +dex increased to 19
|  16   |    35     |  37   |   37   |  18   |
|  17   |    36     |  38   |   38   |  18   |
|  18   |    38     |  40   |   40   |  19   | +3 potency rune
|  19   |    41     |  43   |   43   |  19   | proficiency increased to master for unarmored/light/med
|  20   |    43     |  44   |   44   |  19   | +dex increased to 20 
+-------+-----------+-------+- ------+-------+

As one can see proficiency makes a very significant impact on AC. Even if one were to go with a dex of 10 only at level 1 would this be an improvement over light/ unarmed but that difference would be invalidated by 4th level

+-------+-----------+-------+--------+-------+
| Level | Unarmored | Light | Medium | Heavy |
+-------+-----------+-------+--------+-------+
|   1   |    13     |  15   |   16   |  16   |
|   2   |    14     |  16   |   17   |  16   |
|   3   |    15     |  17   |   18   |  16   |
|   4   |    16     |  18   |   19   |  16   |
|   5   |    18     |  21   |   22   |  17   | +1 potency rune, +Dex increased to 12
|   6   |    19     |  22   |   23   |  17   |
|   7   |    20     |  23   |   24   |  17   |
|   8   |    21     |  24   |   25   |  17   |
|   9   |    22     |  25   |   26   |  17   |
|  10   |    24     |  27   |   28   |  17   | +dex increased to 14
|  11   |    26     |  29   |   30   |  18   | +2 potency rune
|  12   |    27     |  30   |   31   |  18   |
|  13   |    30     |  33   |   34   |  18   | proficiency increased to expert for unarmored/light/med
|  14   |    31     |  34   |   35   |  18   |
|  15   |    33     |  36   |   36   |  18   | +dex increased to 16
|  16   |    34     |  37   |   37   |  18   |
|  17   |    35     |  38   |   38   |  18   |
|  18   |    37     |  40   |   40   |  19   | +3 potency rune
|  19   |    40     |  43   |   43   |  19   | proficiency increased to master for unarmored/light/med
|  20   |    42     |  44   |   44   |  19   | +dex increased to 18 
+-------+-----------+-------+--------+-------+

The only real benefit I can see from this is that at level 1 it frees up some ability boosts away from dex to other stats. Now if one spent their first general feat at 3rd (or possibly 1st if human) on armor proficiency you actually will see an improvement in AC until higher levels (13 for light/medium and 19 for unarmored) where better armor proficiency will supersede heavy armor's initial bonus.

+-------+-----------+-------+- ------+-------+
| Level | Unarmored | Light | Medium | Heavy |
+-------+-----------+-------+- ------+-------+
|   1   |    15     |  17   |   18   |  16   |
|   2   |    16     |  18   |   19   |  16   |
|   3   |    17     |  19   |   20   |  22   | feat for heavy proficiency (trained)
|   4   |    18     |  20   |   21   |  23   |
|   5   |    20     |  23   |   23   |  25   | +1 potency rune, +Dex increased to 16
|   6   |    21     |  24   |   24   |  26   |
|   7   |    22     |  25   |   25   |  27   |
|   8   |    23     |  26   |   26   |  28   |
|   9   |    24     |  27   |   27   |  29   |
|  10   |    26     |  28   |   28   |  30   | +dex increased to 18
|  11   |    28     |  30   |   30   |  31   | +2 potency rune
|  12   |    29     |  31   |   31   |  32   |
|  13   |    32     |  34   |   34   |  33   | proficiency increased to expert for unarmored/light/med
|  14   |    33     |  35   |   35   |  34   |
|  15   |    34     |  36   |   36   |  35   | +dex increased to 19
|  16   |    35     |  37   |   37   |  36   |
|  17   |    36     |  38   |   38   |  37   |
|  18   |    38     |  40   |   40   |  39   | +3 potency rune
|  19   |    41     |  43   |   43   |  40   | proficiency increased to master for unarmored/light/med
|  20   |    43     |  44   |   44   |  41   | +dex increased to 20 
+-------+-----------+-------+- ------+-------+

Note that I did not consider different armors within each type as once you reach full dex cap for your armor the AC differences do not matter, nor did I consider a starting 18 dex for unarmored as that gets into the realm of a very weird barbarian without maxing its key attribute.