I think that a more natural way to write that sentence would be with the use of the possessive determiner your. That definitely makes more sense since we are talking about your human eyes (note that it's not just one eye, but both of your eyes) as opposed to any other types of eyes that there are (animal eyes, for example):
Carrots are good for your eyes.
This is an example of the so-called impersonal you. That's when the personal pronoun you is used to refer to all people in general rather than you personally. I'm pretty sure that you've already seen you used like that many times in English (e.g. Smoking is bad for you. Brushing your teeth is healthy.).
And because we now understand that the eyes which we're talking about are a pair of very specific eyes (our human eyes as opposed to any other types of eyes), it follows that in other to preserve the specificity of this situation, if we wanted to get rid of your, we would have to switch to the definite article the:
Carrots are good for the eyes.
Again, it's not just a single eye, but eyes. Do you think that eating carrots is going to have its effects only on one of your eyes? Only the right eye is going to be affected? Or maybe only the left one? I think both are going to be affected! That's why eyes should be in the plural rather than singular.
The exact same logic applies to your second example (I added more context to the example to make it sound a bit more idiomatic):
I have cut down on drinking too much coffee because my doctor says that drinking a lot of coffee is not good for my teeth.
To be honest, my teeth in this case sounds a lot better than the teeth. Since I'm saying that it is me who has cut down on drinking too much coffee, I'm not talking about your, his or her teeth, but my own teeth! With the teeth, you would have to provide your listener with more information about the teeth that you're talking about:
I have cut down on drinking too much coffee because my doctor says that drinking a lot of coffee is not good for the teeth that are in my mouth.
Wouldn't you say that my teeth sounds more natural and is just a lot shorter to say than the teeth that are in my mouth?