First of all, this is probably a poor idea even if it is legal. Verse, and song lyrics, are notoriously hard to translate well, and make poor examples for language learning, They tend to employ metaphor, allusive language, and idiom heavily, and will in many cases distort the sense of the language for the sake of rhyme, meter, or other auditory effects. But suppose you still want to go ahead.
The developer of such an app could use songs old enough to be in the public domain. "The Star-spangled Banner' for example is not protected by copyright. Neither is the French national anthem, the La Marseillaise. Most well-known songs published before 1900 will not be under copyright. But more recent songs will almost surely be protected.
The use of a line or two from a song might well be permitted under fair use (in the US) or fair dealing (in those countries that recognize this exception to copyright). Use of the entire lyrics of any given song would be more questionable. The use for educational purposes would tend to favor the applicability of any such exception, but the availability of the whole lyric (or even a major part) might serve as a replacement for the original, and harm the market for the original.
No one can ever be sure in advance if a given use will be ruled to be a fair use. That determination depends heavily on the specific facts of each case.
It would avoid risk to use songs that are out of copyright, or short sections of songs, if songs are to be used at all.