I need an counter example showing that $\infty$ is not right exact, that is, an example that for an exact sequence of $G$-homomorphisms of $G$-spaces(not necessarily smooth): $$0\rightarrow U \rightarrow V\rightarrow W\rightarrow 0$$
We have $V^{\infty}\rightarrow W^{\infty}\rightarrow 0$ not exact.
Basically I have trouble even finding a non-trivial exact sequence to work with due to lack of experience, help please.
Any ($\mathbb C[G]$-equivariant) quotient map $V\to W$ gives a short exact sequence $$0\rightarrow U \rightarrow V\rightarrow W\rightarrow 0$$ where $U$ is the kernel. So what you need to show is that $\cdot^{\infty}$ does not preserve epimorphisms.
For any abstract group $G$ there is an epimorphism of left $\mathbb C[G]$-modules $$\begin{align*} \mathbb C[G]&\rightarrow \mathbb C \\ \sum z_g e_g&\mapsto\sum z_g \end{align*}$$ where $\mathbb C[G]$ is the left regular rep and $\mathbb C$ has the trivial action. If you take any non-discrete locally profinite group $G$ then $\cdot^{\infty}$ will map this epimorphism to $0\rightarrow \mathbb C$.