There are many pairs of lovers in this play: Oberon and Titania, Theseus and Hippolyta, Lysander and Hermia, Demetrius and Helena.
As to whether they stay together, the shortest answer is, we don't know. The play ends with the marriage celebration of Theseus and Hippolyta. They are now bound together, but as everyone knows, some couples get divorced.
The most likely answer is, yes, all couples stay together. It is a comedy, after all, and has a happy ending. That said, you can assume that Titania and Oberon will squabble again, sometime in the future.