The paper linked formulates quadratic variation in a measure-theoretic framework. The references therein may also be of interest. As a disclaimer, I did not read this paper very closely, nor is this a research area I am familiar with. I imagine being able to access measure-theoretic tools might offer some interesting approaches to solving this problem. One possible approach that came to mind is as follows.

 In the paper, the authors assign positive finite (Hausdorff) measures to functions of bounded quadratic variation. Their Theorem 20 proves that for every positive finite measure on $[0,1]$, there is a function of finite quadratic variation that generates this measure (and I believe has quadratic variation related to the variation norm of the measure. So a nonzero finite measure would correspond with a nonzero but finite quadratic variation function). It might be true that if the finite positive measure selected is dominated by the Lebesgue measure (assuming this makes sense) then the associated finite quadratic variation function (which may be nonunique?) is differentiable (a.e.?). This is suggested by Theorem 22. Maybe if you choose a finite positive measure that is dominated with respect to the Lebesgue measure with unbounded/discontinuous Radon-Nikodym derivative, you can show the corresponding function of quadratic variation will be differentiable but not continuously so. In which case, there would exist a differentiable function of quadratic variation that is not continuously differentiable with nonzero but finite quadratic variation. This would then prove that the stronger claim ($[f]=0$) in the question is false. This argument is totally speculative and assumes there is some meaningful relationship between the (derivative of the) measure and the derivative of the quadratic variation function. Since such functions with continuous derivatives correspond with the trivial zero measure, it is far from clear what such a relation would be.




The paper: https://www.ams.org/journals/tran/2011-363-08/S0002-9947-2011-05209-8/S0002-9947-2011-05209-8.pdf

Title: HAUSDORFF MEASURES AND FUNCTIONS OF BOUNDED QUADRATIC VARIATION

Authors: D. APATSIDIS, S. A. ARGYROS, AND V. KANELLOPOULOS

Abstract:
To each function $f$ of bounded quadratic variation we associate
a Hausdorff measure $\mu_f$ . We show that the map $f \mapsto \mu_f$ is locally Lipschitz
and onto the positive cone of $\mathcal{M}[0,1]$. We use the measures $\{\mu_f : f \in V_2\}$ to
determine the structure of the subspaces of $V_2^0$ which either contain $c_0$ or the 2
square stopping time space $S^2$.

Chapter 3 is of most interest I think.