I can only come up with one meaning of the word "lines," and it is that the word refers to the lines of the poem, in which the speaker immortalizes his lover's beauty. Beauty and youth eventually fade, just as summer does, and yet he says his lover will enjoy an "eternal summer" because she will "grow" in his "eternal lines." Her beauty and youth are immortalized in his poem, captured forever and unable to fade here. The final couplet helps to strengthen this interpretation:
So long as men can breathe or eyes can see,
So long lives this, and this gives life to thee.
In other words, he says, as long as there are still people living on this earth, "this" will live on, and "this" gives life to thee. To what else could "this" refer other than the poem itself? I cannot think of another plausible interpretation for the word "this," and if "this" refers to the poem, then the "lines" ought to refer to the poem's lines as well. One of the things the speaker dislikes about summer is that its "lease hath all too short a date;" it simply does not last long enough, just like youth and beauty cannot last forever. So it makes sense that the poet, who obviously believes in the value and importance of his words and his work, would pay his lover the ultimate compliment by immortalizing her forever in these lines.