To add a little bit on this.
The original TMNT as of the comic book had masks.
But they were all red as seen in this cover:
Thus they were only distinguishable by the weapons they carried.
The tvTropes page on the franchise gives us even more information.
It seems that the comic was originally in black/white, so the color of the "masks" was not relevant anymore.
But with the first cartoon that changed and the turtles all got their "Color Coded Character" appearance:
Even the special colored issues had the bandana masks they wear all red.
The original cartoon gave the turtles different color bandannas (which have since been used in every successive incarnation) and the action figures, different skin colors (ignored until the 2003 cartoon).
And the tvtropes wiki gives even more information: The Archie comic series Lampshaded this with a "Truth in Television" flashback story about the pre-teenaged Turtles all wearing red headbands and Splinter having difficulty telling them apart until they decide on color coding their masks a la Donald Duck's nephews.
This also seems to be the "in-universe" explanation.
scifi.stackexchange.com/users/31524/Vogel612
To elaborate on part of my comment to Liath's post, the real-world answer is probably just that it was part of the intentional ridiculousness of the "ninja turtle" concept--as described on this page and this one , the idea started as a joke drawing during a sketch-sharing session between creators Kevin Eastman and Peter Laird, you can see that Eastman's first sketch looked even more like a semi-realistic turtle wearing a ninja outfit:
A stubby-legged large-bodied turtle is about the last creature you'd expect to have ninja-like reflexes, hence the humor of giving it a ninja outfit (which naturally includes a mask, since ninjas are known to hide their faces) and a tough-looking expression.
Whether any attempt was ever made to give an in-universe explanation for why they would need masks, I don't know.
You could imagine it was an aesthetic choice on Splinter's part, since as mentioned here his original origin story was that he learned martial arts when he was a pet of a real ninja named Hamato Yoshi and practiced mimicking his moves, so he may have felt motivated to imitate the clothing style as well after he mutated and began to train the turtles.
scifi.stackexchange.com/users/22250/Hypnosifl
If you never heard of this comic and someone were to ask you to draw a "Teenage Mutant Ninja Turtle
" then in all probability you would draw some type of walking turtle wearing a mask and carrying a sword or some other appropriate weapon.
You might also draw it in some kind of ninja outfit.
Point is, in the eyes of the general person, there are certain defining characteristics to what can be called a ninja.
One of them is most definitely the presence of a mask.
Without the mask, it would be "Teenage mutant fighting turtle" or the like that coincidentally might make use of ninja-like weapons.
I doubt the original creator/artist had in mind the practicality of giant turtles masquerading as ninjas.
He just drew the concept as he saw it in his head.
From there, the characters were pretty much established.
scifi.stackexchange.com/users/25439/Carel
So, just to play Devil's advocate here, (fictional) ninja do not wear eye masks.
Doing a google image search for ninja shows tens and tens of images looking like: Their entire body is hidden in black and only the area where the ninja turtles have their masks is visible.
So, how would the ninja turtles have looked like real ninjas?
I drew up a quick sketch:
Doubt anybody would have recognized that as turtles rather than oddly fat ninjas.
So during the first sketching game one of the creators must have decided to instead inverse the entire outfit, leaving some resemblance of how ninja's look without any parts of the actual ninja outfit thus answering why the ninja turtles wear masks instead of ninja outfits.
scifi.stackexchange.com/users/20983/David Mulder
Well, let's see.
Those guys have no eyebrows, so after a couple of minutes of skipping around all that forehead sweat starts going straight into their eyes.
And yes, ok, turtles don't sweat.
But after their transformation they became all sorts of active, so there has to be a way to get rid of all the body heat.
You don't see them walk around with tongues hanging out,
so canine-style heat loss isn't it.
They don't roll around in cool mud, so that's out too.
Basically, they behave like humans.
Most likely they developed an ability to sweat.
So, without eyebrows, headbands are basically a must-have accessory for them.
But they don't exactly have many options of where to put headbands.
Have you seen their heads?
Not a hell of a lot of space up there for a headband.
So their only option is to cut eye holes in the headbands and put them over their face.
It's basic survival for them.
They'd probably gladly take em off, but then they're blind out there.
scifi.stackexchange.com/users/31275/Misha R
The only answer I can give is an out of universe one (I'd love someone to answer in-universe!).
The turtles are colour coded:
Leonardo - Blue Donatello - Purple Michelangelo - Orange Raphael - Red
In the original comics the characters had no voices (clearly)
so their weapons were the only other way to distinguish between the characters, which is no use if they're not fighting or are disarmed.
So the original comic book authors needed some clothing.
What's the distinctive clothing of a ninja?
scifi.stackexchange.com/users/19592/Liath
Ninjas wear masks!
Like in the movie 3 ninjas , they wore masks.
Its also so we can identify them.
scifi.stackexchange.com/users/31548/mystic
This is something I don't understand. Are they trying to hide their identities? How many turtles are there that can stand on their feet? Is it because they are ninjas? Again the shell is a big give away.
Why do the Teenage Mutant Ninja Turtles wear masks?
scifi.stackexchange.com
<teenage-mutant-ninja-turtles>
scifi.stackexchange.com/questions/65415
scifi.stackexchange.com/users/31522/Goozo
Ninjas are known to wear masks as part of their outfit. We also identify the turtles by their masks. The turtles wouldn’t be named Teenage Mutant Ninja Turles without the mask because ninjas wear masks to hide their faces and the mask aids their identity as Ninja Turtles and allows them to mask their faces 
Ninjas are known to hide their faces by wearing masks. The turtles wouldn’t be named Teenage Mutant Ninja Turles without the mask because the mask aids their identity as Ninja Turtles. 