Possessive nouns that end in s can be difficult to navigate, even for those skilled in grammar. The difference between the three choices you listed lie in the punctuation of the word lover. We are choosing whether the correct punctuation for this word (in this context) should be lovers', lovers, or lover's. The correct placement (or lack of) the apostrophe makes up the difference between the choices.

Were we to choose lover's, that would indicate just one person. Since we know that there are two people in this phrase, that is not the correct answer. Lovers is also incorrect because it does not denote possession.

The correct answer in this case is "soul meets soul on lovers' lips." This placement of the apostrophe indicates that both lovers possess their own set of lips. Because lovers is a plural noun, we add an apostrophe but not an additional s.

The correct way to phrase this quotation is "soul meets soul on lovers' lips." In the phrase here, there are two souls and two lovers. This means that the possessive lovers' indicates that there are multiple lovers who own the lips, not just the lips of one lover (which would be stated as lover's lips).

The third phrase is incorrect because there is no possessive apostrophe in the sentence—someone has to possess the lips.

The key to identifying how and when to use apostrophes for possession is identifying the objects and the number of possessors. Is there one person who possesses something? Then use [noun]'s. Are there multiple people possessing an item? Use [noun]'. Are there multiple items, not possessed by anyone? Then use [noun]. Hope this helps!

It's "soul meets soul on lovers' lips" because there are two lovers and two souls, not just one. You must have a second soul to meet the first soul, otherwise it would be "soul meets soul on lover's lips."