What obstacles challenge Lysander and Hermia's love?

There are two significant challenges for Lysander and Hermia to overcome in their love for one another in Shakespeare's A Midsummer Night's Dream.
The first challenge to their love comes from Hermia's father, Egeus, who insists on his paternal right to choose Hermia's husband. Egeus disapproves of Lysander as a suitable match for his daughter and prefers that Hermia marry Demetrius. Hermia refuses to marry Demetrius, so Egeus takes Hermia before Theseus, the Duke of Athens, to have Theseus order Hermia to marry Demetrius.
Hermia asks Theseus to tell her the consequences if she refuses to marry Demetrius.

HERMIA. . . . I beseech your Grace that I may knowThe worst that may befall me in this case,If I refuse to wed Demetrius.
THESEUS. Either to die the death, or to abjureFor ever the society of men. (1.1.64–68)

In other words, "get thee to a nunnery" or die.
At that moment, Hermia chooses the nunnery, but Theseus gives her four days to think it over before she makes her final decision.
In the meantime, Lysander has devised a plan for them to run away together to his aunt's home, which is "seven leagues" away from Athens and thus outside the reach of Athenian law and Hermia's father. Hermia decides that this is a good plan, and she and Lysander agree to run away and meet in the woods, from where they can journey to the home of Lysander's aunt.
The second challenge to their love comes in the form of a love potion that Puck mistakenly puts into Lysander's eyes while he's sleeping in the woods. The effect of the love potion is that Lysander falls in love with the first creature he sees when he wakes up, which happens to be Helena. Lysander falls in love with Helena and falls out of love with Hermia, which leads to a series of merry mix-ups in the woods involving Lysander, Hermia, Demetrius, and Helena.
In time, and after another love potion is administered to Lysander, Theseus finds the two pairs of on-again–off-again lovers in the woods, overrules Hermia's father, and grants his permission for the couples to marry.

THESEUS. Fair lovers, you are fortunately met;Of this discourse we more will hear anon.Egeus, I will overbear your will;For in the temple, by and by, with usThese couples shall eternally be knit. (4.1.178–182)

All's well that ends well.