Okay, these are lyrics, which are poetry, which is art; and the meaning of any piece of art lies within the beholder. But this is my interpretation:
The author of the lyrics is playing around with the word "blue" here, and using it simultaneously for different meanings and even as different parts of speech.
"Blue" can be an adjective, as it is here describing both the color of the girl's eyes (see more on this in @ssav's excellent answer) and the author's sad feelings about being so attracted to this inappropriately under-aged girl (more about this in @Mike Kozar's also excellent answer).
"Blue" can also be a noun; and the same occurrence of the word, meaning both the color blue and the author's sad mood, acts as a subject noun here:
That (article) blue (subject) is getting (verb) me (object) high (adjective)
And (conjunction) [that blue is] making (verb) me (object) low (adjective)
And "blue" can be a verb, but (and this is where your love/hate relationship with the English language may hit a very high/low point) it's not used that way here. Both meanings of "to blue" are chemical processes: one oxidizes the surface of steel to provide partial protection from rust; and one adds a slight blue tint to white laundry, making it the white appear brighter. Neither is going on in this song.
I think the author wants the little girl to close her eyes not just because they are so problematically blue, but also to shut him out, to decline the relationship he so desperately both wants and doesn't want to have with her. Some of the lyrics not quoted here include very sexual connotations and cultural references, and I think that by "eyes" the author might also mean "legs."