There are no rules that allow this, but the DM might allow you to loot a defeated suit or wear an active one with roleplaying consequences.

Note: in 5e the Helmed Horror seems to be a superior, but mostly identical, version of the Animated Armor as referenced by name in the Helmed Horror's entry (MM p.183). After defeat the armor MIGHT be recoverable, as per MM p.15:

You can equip monsters with additional gear and trinkets however you like, using the equipment chapter of the Player's Handbook for inspiration, and you decide how much of a monster's equipment is recoverable after the creature is slain and whether any of that equipment is still usable. A battered suit of armor made for a monster is rarely usable by someone else, for instance.

So it's the DM's call here. Furthermore, upon its "death" a suit of Animated Armor becuse a suit of Inanimate Armor, aka normal armor. So you won't be wearing a suit of Animated Armor per se. Another thing to keep in mind is that a suit of Animated Armor is made out of a suit of full plate armor (as per MM p.19). This costs 1500 gp, which is very high for treasure befitting the treasure you can find for that monster's challenge rating. So the DM should keep that in mind when creating treasure for that level.

Can I wear a suit of Animated Armor?

A suit of Animated Armor is a monster, not an item. Being inside of a monster is frequently the result of being the victim of Swallow Whole, which is not a good thing. And while the Animated Armor is described as being an "empty steel shell" in its entry, the Helmed Horror has got some magical red energy going on. Neither entry describes if you can remove the armor either. While there have been monsters in the history of D&D that improve a host's capabilities when bonding with them, the Animated Armor of 5e is not one of them. So by RAW the answer is No.

However, a player wearing a suit of animated armor might prove to be a fantastic roleplaying oppertunity. Given that the suit and the wearer are not at odds, having a conflict between the will of the user and the orders of the suit can make for some great roleplaying. Perhaps the wearer of the suit is a condemned criminal who was sentenced to wear the suit under the orders of a wizard lord and has to make amends for his crimes? Maybe the suit was send out to find someone or something, and a hapless adventurer put it on and now the wearer has to carry out some objective that does not clash all that much with the suit? This can all be really fun, but you will have to talk this over with your DM (preferably out of game) before you do this.

Yes*.

While none of the rulebooks come out and explicitly say you are allowed to wear robes over your armor (although it may be more realistic to wear your armor over your robes from a practical standpoint), there is circumstantial evidence to support it.

1. In the PHB, the default starting equipment for the Cleric includes both armor (your choice of Chain Mail, Scale Mail, or Leather Armor) and the possibility of clothing (the Priest's pack comes with vestments, and vestments typically include a special kind of robes called a cassock). It would not make sense for a Cleric to be unable to wear his vestments and his armor simultaneously, especially if the Cleric in question takes the particularly active role among his flock that an adventuring Cleric like a PC probably would.
2. As you yourself mentioned, non-magical robes in and of themselves do not provide any protection above and beyond armor, and do not conflict with armor's changes to AC. Even if robes did affect your AC (such as via a magical enchantment), the PHB clearly states that if you have more than one thing which changes the way you calculate your AC, you must choose which one applies.
3. As a consequence of #2 above, the only scenario in which I can imagine there might be potential for abuse is if you are trying to wear magically enchanted robes along with your armor. However, the DMG considers armor to be "Armor" and robes (along with other clothing items) to be "Wondrous Items." No reasonable person would raise the question of whether or not you could, just as examples, wear Gauntlets of Ogre Power with armor, Boots of Elvenkind with armor, or a Cloak of Protection with armor.
4. Expanding on #3, the DMG further has this to say:

Multiple Items of the Same Kind

Use common sense to determine whether more than one kind of a given magic item can be worn. A character can't normally wear more than one pair of footwear, one pair of gloves or gauntlets, one pair of bracers, one suit of armor, one item of headwear, and one cloak. You can make exceptions; a character might be able to wear a circlet under a helmet, for example, or be able to layer two cloaks.

So, even if you don't buy the argument that armor and robes are not the same kind of item (even though they clearly aren't, despite covering the body in largely a similar fashion), the rules provide for the possibility that they can still be combined if your DM chooses to allow it.

Olorin's answer provides further evidence that this is a valid approach to your particular situation, and references a statement of opinion given by one of 5th Edition's lead designers: Mike Mearls. It is important to note, however, that the official arbiter of the rules-as-written is Jeremy Crawford. While the words of Mearls carry significant weight, they are still merely his personal opinion on how the game works at his table.

*The Standard Caveat: D&D 5th edition empowers the DM in ways that 3rd, 3.5, and 4th did not. While rule zero has always applied, 5th Edition chooses not to explicitly codify many things. If your DM says you can't, you can't.