If a PC has plate armor, Str 19, and dons a Cloak of the Manta Ray, can they swim 60 ft. as the DMG says (p. 159), or do they suffer penalties for swimming in heavy armor even with a magically-granted swim speed?
You can do whatever you like; what I would do is play it as written.
PHB p. 144
If the Armor table shows “Str 13” or “Str 15” in the Strength column for an armor type, the armor reduces the wearer’s speed by 10 feet unless the wearer has a Strength score equal to or higher than the listed score.
As it says, if they are strong enough they suffer no penalty to speed, if not then they lose 10 feet of speed.
PHB p. 182
Climbing , Swimming , and Crawling
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed. ... Similarly, gaining any distance in rough water might require a successful Strength (Athletics) check.
As it says, a creature without a swim speed uses 2 feet of movement for every 1 foot covered (3 if difficult terrain). A creature with a swim speed moves 1 for 1 (2 for difficult terrain). As phrased, a swim speed does not exempt you from making a Strength (Athletics) check if required.
Significantly, it says nothing about the effects of armor on swimming.
Does this make a lot of sense? From a game perspective, yes - it gives simple rules that can be used with a minimum of complexity.
Can you rule that armor should make a difference? Of course, it says you can right on p. 6 of the PHB - if the PC says "I swim across in my Plate armor" then a perfectly sensible ruling is "You sink like a stone", however, given that this is not what the rules say, fairness dictates that you tell the player the consequence of their action before they undertake it and also before they make a commitment to spend in game money on an expensive anchor.
DMG p. 159
CLOAK OF THE MANTA RAY
Wondrous item, uncommon
While wearing this cloak with its hood up, you can breathe underwater, and you have a swimming speed of 60 feet. Pulling the hood up or down requires an action.
As it says, it gives you a swim speed - take that and apply it to the rules above.
"can they swim 60 ft as the DMG says (pg 159)"?
"are any of you DMs ruling that there are some penalties to swimming in heavy armor even with a swim speed?"
I cannot speak for "any DM", but by the rules in the book: There are no penalties to swimming in armor with or without a swim speed.
"Would you include a higher Str requirement with swimming?"
Me? No, by the rules in the book: The cloak gives you a swim speed of 60 feet. If you meet the strength requirement, there is no penalty to your speed; if you do not then your speed is reduced by 10 feet (i.e. to 50 feet).
There's no rulings making it sink or swim (ha, joke) with armor. However, you're not just carrying armor. Assuming you actually keep all your stuff strapped to your back and in backpack, you've got quite a bit of dense weight (metal armor) and voluminous weight (everything in the backpack). That technically would make you have to try to swim up while swimming forward and because of volume as impediment to forward, and weight as impediment to upward, it'd take a lot more effort.
Think of it this way. it makes sense. the DM is right. Rule 1 or 2 or 3 or whatever. the DM being right essentially means nearly all games are houseruled, that games which directly change the built in rules are houseruled (like all adventurers have infinite underwater breath), or that any DM rulings that are NOT part of the rules of the game in basic, but seem to be rational, are not houserules.
You can rule that people can swim and aren't affected. You can rule that people sink like a stone or have penalties or checks to swimming due to weight.
The thing that swim rules dont make clear is whether those swim rules are for naked monsters and people or for fully weighted steel adventurers or monsters. Since it does not clarify, the common sense approach is that this is for naked.
The rules are filled with massive gaps all over the place for each and every situation. The rules should therefore be seen as "this is how it specifically works WITHIN common sense". As such, it being common sense that its far easier to swim naked than with clothes on ESPECIALLY shoes (try it), there must be something regarding swimming.
When you become some 24 strength godly being, yeah, you swim with no penalties because gravity hasn't got nothing on you! But when you're little above some peasant, with just a little extra muscle, maybe you have to roll for success.
This incidentally makes suffocation/drowning rules matter more, it makes exhaustion rules matter more, it makes don and doff rules matter more.
You should be creatively looking for ways to make the current rules, which make the game more difficult and more DIVERSE, matter more, by ruling checks or possibly sinking while wearing armor.
Think of it like this. If you rule that people can just swim forever, then people can WoWize and just swim across 20 mile lakes in full armor. Nothing to think about. No weighing of risks. That lake might as well not even be there.
How about being in full armor with backpack on a ship in the ocean? WoWize it, and you can sit in your steel cage all safe from passing pirates, air attacks, whatever, because you don't have to make the choice of protecting yourself from potential drowning by staying "light" in regard to your weight.
I think so many dnd "rule holes" would've been easy for DMs to common sense rule if the first two pages of the DMG/PHB said "use some damn common sense" and listed out how physics in the game world worked as long as magic is not part of the equation.