Classes designed around heavy armor [paladins, clerics] will languish. They will be playing catch-up the entire game, until finally, at level 20, they have the AC of a level 1 character in most games.
Classes designed around finesse or ranged weapons, on the other hand, will start with higher AC, and will only increase as they gain levels. Eventually they'll have access to an AC no character can attain in the game without magic.
Dexterity becomes a God Stat, Strength becomes a dump stat.
As for monks and barbarians, just remove the class feature and that's all you need to do. The proficiency bonus is replacing Wis/Con for them so they're not losing out. There's no reason to give them additional defensive benefit for no reason.
An alternate solution: Have each class have a class feature, like the Monk and Barbarian, that can give them AC instead of the proficiency bonus. Fighters NEED to have AC to do their job, so having their AC be 10 [Str or Dex]+Con modifier would be fair. Rangers, on the other hand, do not, so they'd have Str or Dex for their AC.
Basically, rewrite each like this:
Heavy Armor Proficiency: This character can calculate her AC using 10+Constitution modifier+Strength modifier.
Medium Armor Proficiency: This character can calculate AC using 12+the higher of Dexterity or Strength
Light Armor Proficiency: This character can calculate AC using 12+their Dexterity modifier.
Shield Proficiency: This no longer exists. Classes that give it, instead, give Parrying Proficiency. At the beginning of each round, if a character is wielding a one-handed weapon in one hand but the other hand is empty, that character gets a +2 bonus to AC. The character ceases parrying if they activate the versatile trait on their weapon to use it in two hands, or if they wield a two-handed weapon.
References in class features, feats, etc to using a shield are replaced with parrying. For example, a monk who parries cannot use their Unarmored Defense feature.