A wizard wearing armour that they are not proficient in is unable to cast spells. Would it be possible to force a wizard into armour using a combination of Illusion and Illusory Reality? For example, cast an illusion of armour over an enemy wizards body and then make it real for one minute using Illusory Reality:
Illusory Reality. By 14th level, you have learned the secret of weaving shadow magic into your illusions to give them a semireality. When you cast an illusion spell of 1st level or higher, you can choose one inanimate, nonmagical object that is part of the illusion and make that object real. You can do this on your turn as a bonus action while the spell is ongoing. The object remains real for 1 minute. For example, you can create an illusion of a bridge over a chasm and then make it real long enough for your allies to cross.
The object can't deal damage or otherwise directly harm anyone.
If possible, this should prevent the wizard from then casting any spells. I understand that this may be up for DM discretion. However my question is, as RAW is this possible? Or would this be classed as 'Harming' the wizard from a RAW point of view?
Is harm purely defined as dealing any amount of damage? Or does it also fall into the realm of causing issues for opponents?
Given that, as you say, the illusion of armour, then the armour being made real, do not 'harm' the enemy, then there's no reason why this would contradict the emboldened part of your quote: "The object can't deal damage or otherwise directly harm anyone".
The fact that the enemy wizard is now unable to cast spells is simply a side effect. Even if it is considered harm, it's indirect harm at worst, which still doesn't contradict the emboldened text.
Since this is your wizard's 14th level archetype feature, finding ways for it to be comparable to the other archetype's 14th level abilities doesn't make this an overpowered use of the ability either. Consider the other archetypes, such as a Conjuration wizard's "Durable Summons" (PHB, pg. 116):
Starting at 14th level, any creature that you summon or create with a conjuration spell has 30 temporary hit points.
Or an Evocation wizard's "Overchannel" (PHB, pg. 118):
Starting at 14th level, you can increase the power of your simpler spells. When you cast a wizard spell of 5th level or lower that deals damage, you can deal maximum damage with that spell.
As you can see, some of these are more directly useful in combat, so if you've thought of a way to make "Illusory Reality" keep up with them in combat, good for you! (I'd allow it if I was your DM!)
Note that ultimately your DM has the final say as to whether this would work, and if your DM just so happened to make his NPC wizard proficient with whatever armour you put on them (since a DM can adjust the stats of any creature, or you could justify it by claiming that the wizard is multiclassed into something with armour proficiencies), then this wouldn't work either (but at that point, they're doing so to deliberately undo your plan, in which case they're better off simply ruling that it doesn't work).
This is in grey area of "harm" and open to DM interpretation and ruling.
In terms of game balance and conceptually, this is not different to having an illusion of locked manacles on almost any character. There are two issues with it:
Casting an illusion that syncs perfectly with another, possibly active opponent, character.
Almost arbitrary control of restraint-like effects provided there is no direct physical damage. For instance, there is no requirement that an illusion of heavy armour has straps etc making it possible to take off. Or joints that move. How about a gag, or a difficult-to-remove helm with no eye holes . . .
For (1) I would suggest that you would need either a willing target for low-level spells, or a spell description that makes it clear that the created items are worn or may be in sync with the target. Depending on spells available in the campaign, this might make the situations in (2) more reasonable.
Spells the armour idea would work on, in my opinion:
Disguise Self (a clear limitation there, but perhaps interesting in reverse if you really want some armour, and don't wish to cast spells)
Seeming I think this would do exactly what you want. An unwilling target would get a Charisma saving throw, although you might be able to trick someone into thinking they were getting a disguise, than make it real in order to disable their spellcasting.
As an added bonus with Seeming, you can make allies look like enemies and enemies look like allies, or either look like innocent bystanders, giving your team an advantage due to confusion. Adding a rider due to the 14th-level school ability that almost completely disables one of your enemies is very nice.
I have to challenge the premise of the question:
A wizard wearing armour that they are not proficient in is unable to cast spells.
The question does not specify is the wizard PC or NPC, so I assume both cases.
For PC, there is a paragraph in the Player's Handbook:
Your class gives you proficiency with certain types of armor. If you wear armor that you lack proficiency with, you have disadvantage on any ability check, saving throw, an attack roll that involves Strength or Dexterity, and you can't cast spells.
So yes, if a PC wizard lacks armor proficiency, he/she can't cast spells being covered in armor.
For NPCs there is no such rule. In 5e, NPCs have no explicit armor proficiency described in their stat blocks, neither they have classes. Moreover, there are spellcasters in MM that wear armor (a Death Knight, for instance). Therefore, in terms of mechanics, NPC caster interactions with armor is entirely up to the DM.
Turn-based mechanics could make an impression that enemies stand still all the time, except of their respective turns. However, from the narrative perspective, there is no turns (or turns are simultaneous, so to say), so there is no "not my turn" moments at all. That means you have to latch the illusion to a moving target somehow. As far as I know, there is no spells capable of this. You can create an illusion of a standalone item, you can even move this illusion afterwards, but you can't create illusion of this item being worn by your enemy (which is, by the way, fighting with you right now).
I have to emphasize that there is no condition that prevents spellcasting in 5e. The only (kind of) exception is Incapacitated, which prevents all actions. The 5e way to shut down casters is to restrict them them from using spell components, not the very spellcasting feature.
Even if there was such a condition, the spell description states:
The object can't deal damage or otherwise directly harm anyone.
So the object not only can't deal damage, but it also can't "harm". "Harm" is not a strict term in 5e, but I guess that means any bad conditions a character can face. That includes blinded, deafened, restrained, incapacitated, etc. Preventing spellcasters from using their magic in a forcible way is "harm".
I'm going to be cantankerous here for a moment since everyone else is being so reasonable in allowing it.
Casting spells is central to a wizard, but as stealth is central to a Rogue, fighting is to a Fighter, or halving is to a Halfling. I would suggest that a spell or effect that removes any class's ability to "be that class" is in fact harm.
It's actually a collection of several parts. Chain mail is hundreds if not thousands of links. A breast plate is a metal plate plus fitted leather armor. Leather armor is itself at least two pieces (pants + shirt).
Unless the wizard is asleep/frozen/held in place, the armor is going to have to be animated to move along with the wizard otherwise it would be a pretty shoddy illusion, AND not actually be being worn at the time. Remember a turn is 6 seconds and technically everyone's turn in a round happens at same time, the other PCs and NPCs aren't frozen in place during your turn, that's just a conceit to make the game flow better.