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I'm looking for a simple example of a compact Lie group $G$ with a maximal torus $T$ such that $G$ acts on some manifold $X$ non-trivially but the action of $T$ on $X$ (defined by restricting the $G$-action) is trivial.
Is there an example of this? Or if not, how would one prove it? Thanks.
If $G$ is connected, you can not have such examples. Recall that in a connected compact Lie group $G$ any element is conjugate to an element of $T$. So given $g \in G$ and $ x \in X$ we have $g = hth^{-1}$ for some $h$ in $G$ and therefore
$$ g x = hth^{-1}x = h t (h^{-1}x) = hh^{-1}x = x $$
That is, the action of $G$ must be trivial.
I will give you now an example where it happens.
Consider the usual maximal torus $T$ of $SU_2$, that is, the subgroup of diagonal matrices. In particular $T=S^1$ is a manifold. Let $G$ be the normalizer of $T$ in $SU_2$, which is a compact Lie group with $T$ as a maximal torus. Since $T$ is normal in $G$, we have an action of $G$ on $T$ by conjugation. Consider the element
$$ x = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) \in G$$
Then we have
$$ \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) \left( \begin{array}{cc} z & 0 \\ 0 & z^{-1} \end{array} \right) \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) = \left( \begin{array}{cc} z^{-1} & 0 \\ 0 & z \end{array} \right) $$
So the action of $G$ on $T$ is not trivial. However the restriction of the action to $T$ is trivial, because $T$ is commutative. Of course the point here is that $N$ is not connected.
