As explained here Converse of Itô's formula, such a formula would imply that f is C2-differentiable

Let $f,h,g$ be continuous functions and $B$ a real Brownian motion. We suppose that a.s. $$\forall u \in \mathbb{R}_+,f(B_u)=f(B_0)+\int_0^ug(B_r)dB_r+\frac{1}{2}\int_0^uh(B_r)dr.$$ Prove that $f$ is a $C^2$-function, $f'=g$ and $f''=h.$

In terms of semi-counterexample, the absolute value function is uniformly contiuous but it satisfies a formula that requires the addition of Local time (and so having an Ito formula would imply that local time is zero) https://en.wikipedia.org/wiki/Tanaka%27s_formula

$$|B_t| = \int_0^t sgn(B_s)\, dB_s + L_t.$$

There are weaker versions that require only convexity eg. The Ito-Tanaka-Meyer Formula.

As explained at Converse of Itô's formula, such a formula would imply that $f$ is $C^2$-differentiable.

Let $f$, $h$, $g$ be continuous functions and $B$ a real Brownian motion. We suppose that a.s. $$\forall u \in \mathbb{R}_+,f(B_u)=f(B_0)+\int_0^ug(B_r)dB_r+\frac{1}{2}\int_0^uh(B_r)dr.$$ Prove that $f$ is a $C^2$-function, $f'=g$ and $f''=h$.

In terms of semi-counterexample, the absolute value function is uniformly contiuous but it satisfies a formula that requires the addition of Local time (and so having an Itô formula would imply that local time is zero). Tanaka's formula:

$$|B_t| = \int_0^t \operatorname{sgn}(B_s)\, dB_s + L_t.$$

There are weaker versions that require only convexity e.g. The Itô–Tanaka–Meyer Formula.