Using exponent laws, the expression $\frac{2^{x+y}}{2^{x-y}} = 2^{(x+y)-(x-y)} = 2^{2y}$. Since $x$ and $y$ are positive integers with $xy = 6$, then the possible values of $y$ are the positive divisors of 6, namely $1, 2, 3$, or 6. (These correspond to $x = 6, 3, 2, 1$.) The corresponding values of $2^{2y}$ are $2^{2} = 4, 2^{4} = 16, 2^{6} = 64$, and $2^{12} = 4096$. Therefore, the sum of the possible values of $\frac{2^{x+y}}{2^{x-y}}$ is $4 + 16 + 64 + 4096 = 4180$.