It may happen that
$$\dim_H X/G<\dim_H X-\dim_H (G\cdot o);$$
the following example is almost identical to Example 7.8 in "Fractal geometry" by Kenneth Falconer which provides spaces $X$ and $Y$ such that
$$\dim_H (X\times Y)>\dim_HX+\dim_HY.$$

Denote by $W(\varepsilon)$ the two-point space with distance $\varepsilon$ between the pair of points.
Consider the $\ell_\infty$-product
$$X=\prod_{n\in\mathbb{N}}W(\tfrac1{2^n}).$$
Fix a subset $\Sigma\in \mathbb{N}$ and consider the natural action of group
$$G=\prod_{n\in\Sigma}\mathbb{Z}_2$$
on $X$.

The orbit can be identified with the infinite product
$$G\cdot o=\prod_{n\in\Sigma}W(\tfrac1{2^n}).$$
and the quotient space can be identified with the infinite product
$$X/G=\prod_{n\notin\Sigma}W(\tfrac1{2^n}).$$

Note that for appropriate choice of $\Sigma$, we have
$$\dim (G\cdot o)= \dim X/G=0$$
while
$$\dim X=1.$$