Index of Modified Dirac Operator

For this particular case the two operators are conjugate (though note the conjugacy is not unitary unless $s$ is imaginary)

$$ D_{f,s}=e^{-sf}D e^{sf} $$ so that the dimension of both the kernel and cokernel are independent of $s$. Note that Chris Gerig's comments on the other hand apply more generally. The index of $$ D_{s,\theta}=D+s\theta⋅ $$ for any one-form $\theta$ is independent of $s$.

Making my comment a formal answer: The perturbation object is a compact operator (for any scaling $s$), and $D$ is Fredholm. The space of Fredholm operators is open in the (Banach) space of bounded linear operators, and moreover $ind(D)=ind(D+K)$ for any compact operator $K$.
The question about spectral flow (jumps in the kernel) is much more intricate, it should depend at least on the critical points of $f$ and the magnitude of $s$. But again by general facts of Functional Analysis, for $|s|$ sufficiently small the dimension of the kernel won't jump (and in general it will only jump for a discrete set of $s\in\mathbb{R}$).