Can you wear a robe over armor?

Question

More specifically: Can you wear a robe of eyes over plate armor? Does it say so in the rules? I thought you can if the other clothing does not provide any AC.

Answer 1

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.

Answer 2

Yes

The rules

Armor and magical robes are different categories of items, so they can be worn freely together. While plate armor is categorized as Armor, robes are classified into the catch-all category of Wondrous items (DMG p. 140).

However, should the robes give you a different way to calculate your AC (like the Robe of the Archmagi - DMG p. 194), you have to choose which to use (the sane choice being the one that yields the highest result).

Armor Class (PHB p. 14)

Some spells and class features give you a different way to calculate your AC. If you have multiple features that give you different ways to calculate your AC, you choose which one to use.

The wisdom of the sages

Mike Mearls voiced his opinion on this matter (coincidentally, exactly regarding wearing the Robe of eyes over platemail) in a sage advice:

charles ‎@alfabetsoupe

@mikemearls Can magic items such as Robe of Eyes be worn over armor, such as Platemail? Or does the robe occupy a sole "torso slot"?

Mike Mearls ‎@mikemearls

@alfabetsoupe you can wear it - only limit is attunement cap