The entropy of a noisy distribution
Such a bound is not possible. Consider the case where $f$ is the distribution that is uniform over some set $S$ of size $2^{\delta \cdot n}$, and let $\tilde{f}$ be the distribution that with probability $\delta$ outputs a uniformly distributed element of $S$, and otherwise outputs a uniformly distributed string.

It is not hard to see that you can get from $f$ to $\tilde{f}$ you only need noise of at most $(1-\delta) \cdot 2^{-\delta \cdot n}$.  However, $H(f)= \delta \cdot n$ while $H(\tilde{f}) \approx (1 - \delta + \delta^2) \cdot n$. Thus, you get a difference of $(1 - \delta)^2 \cdot n$ for arbitrarily small $\delta$ for an extremely low noise.

In particular, you can set $\delta = \frac{\log(1/\varepsilon)}{n}$, and obtain noise $\varepsilon$ and entropy difference $\approx n - 2\log(1/\varepsilon)$.