I'll focus on potential topological characterizations of the action. As I mentioned in the comments above, every element of $G$ will either lie in a maximal compact subgroup of $Aut(\partial{T})$, or will have north-south dynamics (a hyperbolic element).

If the group $G$ acts discretely, cocompactly on $\mathcal{T}$, then there's Bowditch's characterization that this is equivalent to $G$ acting properly dicontinuously and cocompactly on the triple points in $\partial{T}$. In fact, in this case $G$ will be a virtually free hyperbolic group, and a (finite) graph of finite groups.

I believe that $Aut(\mathcal{T})$ is closed in $Aut(\partial{\mathcal{T}})$, with the induced topology. In this case, one may assume that $G\leq Aut(\mathcal{T})$ is closed, otherwise taking its closure $\overline{G}$ in $Aut(\partial{\mathcal{T}})$ will correspond to taking its closure in $Aut(\mathcal{T})$, and if $\overline{G}\leq Aut(\mathcal{T})$, then $G\leq Aut(\mathcal{T})$. So assume now that $G$ is closed in $Aut(\partial{\mathcal{T}})$.

If $G$ is a compact subgroup of $Aut(\partial{\mathcal{T}})$, then I think it should
be a profinite group, which acts on a tree (elliptically with a global fixed point).
However, I'm not sure exactly how to tell if the action of $G$ on $\partial{\mathcal{T}}$
corresponds to this group action. Let's assume from now on that $G$ is not compact.

If $G$ does not act cocompactly on $\mathcal{T}$, then consider the limit set $\Lambda(G)\subset \partial{\mathcal{T}}$, which is the set of accumulation points of $Gx\subset \partial{\mathcal{T}}$ for any $x\in\partial{\mathcal{T}}$. One may realize this as the closure of the limit points of
hyperbolic elements of $G$. Then $G$ should act cocompactly on the convex hull of $\Lambda(G)=\mathcal{R}$ inside of $\mathcal{T}$. So one may replace $\mathcal{T}$ with $\mathcal{R}$, and $\partial{\mathcal{T}}$ with $\partial{\mathcal{R}}=\Lambda(G)$.

So now assume that $G$ is a closed noncompact subgroup of $Aut(\partial{\mathcal{T}})$, and $\Lambda(G)=\partial{\mathcal{T}}$.
Now, I believe that there should be a generalization of Bowditch's theorem. $G$ should be hyperbolic as a topological group, generated by a compact subset. I think this should be equivalent to $G$ acting properly cocompactly on the triple point space of $\partial{\mathcal{T}}$, but I don't know if this is proven (or even correct). Compare to the action of $Isom(\mathbb{H}^n)$ on $(\partial\mathbb{H}^n)^3-\Delta$, which is proper and cocompact.

There is another possible topological characterization in terms of actions on spaces of walls, but this is really just an encoding of the tree on $\partial{T}$ in terms of how each edge partitions $\partial{T}$ into pairs of clopen sets. If $G$ is closed, but noncompact and nondiscrete, then one might be able to encode the tree by the maximal compact subgroups of $G$ and their intersections, essentially Bass-Serre theory. But I haven't thought this through.