If it's made by dwarves, we call it dwarven. If it's made by elves, we call it elven. Just because it's called Dwarven Plate does not mean it's made only for dwarves.

For magic items (which is what Dwarven Plate is), page 140 of the DMG states;

In most cases, a magic item that's meant to be worn can fit a creature regardless of size or build. Many magic garments are made to be easily adjustable, or they magically adjust themselves to the wearer.

Rare exceptions exist. If the story suggests a good reason for an item to fit only creatures of a certain size or shape, you can rule that it doesn't adjust. For example, armor made by the drow might fit elves only. Dwarves might make items usable only by dwarf-sized and dwarf-shaped characters.

And as Quentin states, for general rules on armour there's page 144 of the PHB;

In most campaigns, you can use or wear any equipment that you find on your adventures, within the bounds of common sense. For example, a burly half-orc won't fit in a halfling’s leather armor, and a gnome would be swallowed up in a cloud giant’s elegant robe.

The DM can impose more realism. For example, a suit of plate armor made for one human might not fit another one without significant alterations, and a guard’s uniform might be visibly ill-fitting when an adventurer tries to wear it as a disguise.

There's also nothing in the item's description that restricts it to only dwarven characters. Unlike the Dwarven Thrower which specifically says it needs to be attuned to a dwarf.

So, as a general rule, whether found or custom made, (within reason) it can be worn by any race.
It's then up to you to decide what limitations/realism you add to it I.E if it's found, does it auto adjust to fit non-dwarves or would they have to find someone to adjust it for them, are the dwarves actually willing to make this very rare magical armour for other races without good reason.

One thing to remember though is be consistent. If you put restrictions on one piece of armour (magical or not) you should also apply restrictions to others.

Another thing to consider is to ask the other players (and DM if you're not DMing) what level of immersion they want.
Do they want to simply be able to wear any armour they come across or do they want that extra bit of realism that makes them drag their loot half way across the country to find someone willing the make the necessary adjustments.

The Rules Don't Say

As nvoigt pointed out, the PHB describes chain mail and plate mail as complete items. They are either on, or they're off. There are rules for how long it takes to put them on and take them off, but there is no description for your AC while wearing just the padding.

Furthermore, the padding under plate and chainmail isn't padded armor. Padded armor is a different thing. When you buy plate mail you don't get plate mail armor that has padded armor inside, you get plate mail armor that has padding inside.

You Could Try It

Since there are no rules that describe this situation, you are down to the most general rule of all. At the beginning of the Players Handbook, it says:

1.The DM describes the environment. ... 2.The players describe what they want to do. ... 3.The DM narrates the results of the adventurers' actions. (PHB p.6, "How to Play")

So, your character could just try it and see what happens. "Try it" could mean a lot of things. "My character takes off all his armor but the padding. What's his AC?". "I'm looking at the swamp we're about to cross. I'm an experienced fighter, I've been wearing armor for years, I've been trained by experts. I know if I fall in wearing full plate, I'll probably drown before I can get out of it. What does my training tell me about whether I can wear just the padding in order to mitigate the drowning risk but still retain some protection?" Or perhaps your armor wearer might consult armor experts.

You've Solved My Drowning-in-Armor Puzzle!

As a GM, I view the water-armor situation as a challenge or problem for the PCs. Which do you do, take off the armor and carry it thereby lowering your AC, or wear it and risk drowning?

Only it isn't an either-or, which is the beauty of RPGs. There are other solutions to the problem. Perhaps the water is such and the PC's strength is such that the PC can just walk out, perhaps with a strength or constitution check. Perhaps the PCs have access to magic that allows them to breathe water. Maybe they can bypass the swamp. The players might (and hopefully will!) have innovative solutions I haven't thought of.

If the PC said, "I take off most of my armor, and wear only the padding", or "only the helmet, gauntlets, greaves and leg guards" and subsequently fell into the water I think the time to remove the armor and thus the risk of drowning would be less, while the chances of swimming in part of the armor would be more. And if the PC were attacked by Rodents of Unusual Size while crossing the swamp, I'd definitely have to allow that the PC's base AC in padding or other bits and pieces was greater than without armor. Wearing only part of the armor might be a reasonable solution to the but-I-might-drown problem. Although wearing only the padding might be really bad for it, given briars and such.

Given that I don't want to play "Armor & Anoxia" I'd be reluctant to come up with a codified houserule that says you can wear half the armor for half the AC and half the risk of drowning, but perhaps that would be the solution. I'd definitely want to keep things liquid, and not weigh the game down with heavy armor questions, but I'd definitely want to pad it enough to keep things moving forward, while at the same time rewarding innovative role-play and problem-solving.