Is $\mathbb{R}^{\infty}$ homeomorphic to $\mathbb{R}^{\infty}\setminus\{0\}$?

The answer is yes.

In Chapter VI § 2 of Selected Topics in Infinite Dimensional Topology (Bessaga & Pełczyński) is stated the surprising result that $\mathbb{R}^\infty$, the Hilbert space $\ell_2$ and the unit sphere $S \subset\ell_2$ are homeomorphic spaces (${}^*\!$). The background required (developed in the previous chapters) seems to be rather wide. Anyway, proving this result, they exhibit an explicit homeomorphism between $S \smallsetminus \{\ast\}$ and $\ell_2$.

(${^*}\!$) moreover there is a 1966 paper by Bessaga entitled Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere

In "Infinite-dimensional Topology, Prerequisites and an introduction" by Jan van Mill (table of contents) the author proves a characterization of $R^\infty$, also called $s$ in the book, which allows him to show that for any $\sigma$-compact subset $A$ of $\mathbb{R}^\infty$ we have that $\mathbb{R}^\infty \setminus A$ is homeomorphic to $\mathbb{R}^\infty$. There is considerable machinery involved, but it's a very elegant exposition of these matters, IMHO. So the infinite product of copies of $\mathbb{R}$ behaves quite differently from its finite powers, in which homological/homotopical methods can be used to show e.g. $\mathbb{R}^n$ and $\mathbb{R}^n \setminus \{p\}$ are not homeomorphic.