How can I linearize the IFF-THEN constraint with binary and continuous variable?

First, note that you can't impose strict $>$ constraints. Instead, you'll have to choose a small $\epsilon > 0$ and then require $f \ge \epsilon$. Then these constraints should do the trick:

$$\begin{align} f & \le Ma \\ f & \ge \epsilon - M(1-a) \end{align}$$

Here, $M$ is a large constant. You should set it equal to the largest value that $f$ can reasonably take.

The logic is: If $a = 0$, then the first constraint says that $f$ must equal $0$ (and the second constraint has no effect). If $a = 1$, then the second constraint says that $f$ must be greater than or equal to $\epsilon$ (and the first constraint has no effect).