What is the sum of the solutions to the equation $\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}$?
Let $y = \sqrt[4]{x}$. Then we have
$y(7 - y) = 12$, or, by simplifying,
\[y^2 - 7y + 12 = (y - 3)(y - 4) = 0.\]
This means that $\sqrt[4]{x} = y = 3$ or $4$.
Thus the sum of the possible solutions for $x$ is $4^4 + 3^4 = \boxed{337}$.