I assume by *Lebesgue differentiation theorem* you mean the statement that $|B(x)|^{-1}\int_{B(x)} f(y)\, dy \to f(x)$.

Then it's not clear to me what your set-up on $X=[0,1]^{\mathbb N}$ is (what's the measure?), but in any event, for an *arbitrary* metric, this already fails on $\mathbb R^2$. You can take a metric that gives you wide thin rectangles as small balls, for example
$$
d(x,y)=\max (|x_2-x_1|, |y_2^{1/3}-y_1^{1/3}|)
$$
(if $y<0$, then $y^{1/3}$ just means $-|y|^{1/3}$).