Problems in the performance, and not related to themes? The most immediate of these issues are practical. First, creating the disguises. You have in As You Like It Rosalind, a young woman who is supposed to be attractive; you have to make her recognizable as a woman to the audience, but at the same time not recognizable to the other characters. Likewise, Prince Hal has to disguise one of the most famous faces in the camps (probably the most famous) to move among the men. We need to realize it is Hal, but no one else can.
Closely related to this is the problem of suspension of disbelief. Though the emotions are very different (more comic for Rosalind), you have to convince the audience to not start doubting that the men believe.
A subtler problem is that their true emotions and characters must show through. This is a big challenge for the performers. Rosalind must show and conceal love at the same time; Hal must show and conceal kingship.