Here is the wording of the lyric that appears in Edmund Fellowes, English Madrigal Verse, 1588–1632 (1920):
Though Philomela lost her love,
Fresh notes she warbleth, yet again.
He is a fool that lovers prove,
And leaves to sing to live in pain.
The main differences between this version and the one that the poster cites are in line 2: "notes" for "note," and "yet" for "yes!" There are also some differences in punctuation—most notably the period at the end of line 2 in Fellowes's version as opposed to the semicolon in the poster's version.
The period ending line 2 is significant, I think, because it supports the suggestion that Spagirl makes (in a comment above) that there may be no connection between the "He" in line 2 and any character in the Philomela myth to whom lines 1 and 2 may allude.
On the face of it, the song first offers two lines about the behavior of the nightingale (imagined as a transformed girl), which sings though she is alone and lovelorn. Then it turns to consider a person (I nominate the songwriter) who has chosen (perhaps) to test the love of two lovers (I nominate himself and his beloved) in some unstated way; and finally it accompanies that person as he leaves the presence of the lover[s] to sing alone in pain—like the anthropomorphized nightingale.
The particular line of interest—"He is a fool that lovers prove"—may use the verb prove in the now semi-archaic sense of "test" that tchrist points out in a comment above (as my initial reading of it does). But another possibility involves an even more out-of-the-way meaning of prove, which Merriam-Webster's Eleventh Collegiate Dictionary (2003) gives as follows:
prove ... vt (13c) 1 archaic : to learn or find out by experience
This meaning of prove is the one that Marlowe has in mind in "The Passionate Shepherd to His Love" (written no later than 1593):
Come live with me and be my Love,
And we will all the pleasures prove
That hills and valleys, dales and fields,
Or woods or steepy mountain yields.
Marlowe isn't offering to establish by ratiocination that such pleasures exist; he is urging his love to experience them with him.
This sense of prove invites a different interpretation of line 3—something like
He is a fool who learns first-hand the ways of lovers
—namely that they will break your heart, and you will find yourself, like Philomela, flitting from tree to tree as the night deepens, singing your song of sorrow and hard-earned wisdom.
Either reading requires a little help from the listener/reader to make perfect syntactical sense. Specifically you have to supply a dropped auxiliary verb:
He is a fool that lovers [doth] prove,
Without the (arguably) missing doth, you have a statement that seems syntactically straightforward ("He is a fool whom lovers test [or learn by experience]") but doesn't stand up to logical scrutiny (why does a lover's decision to test you or experience you make you a fool?).
But once you interpolate the doth, the singer's complaint makes sense—and you get to choose between interpreting the lyric as a lament for having needlessly and foolishly put one or more lovers to the test, and interpreting it as a lament for having been victimized by foolishly learning through bitter experience about lovers and their deceits. On that front, my advice is "When you're dealing with a songwriter, never bet against the more self-pitying meaning."