One set of sufficient conditions may be obtained if your map plays nicely with respect to subspaces fixed by closed subgroups. Let $X$ and $Y$ be $G$-spaces for any $G$ (not only the torus) and assume that you have an equivariant map $f:X \to Y$. If for any closed subgroup $H < G$ the induced map $X^H \to Y^H$ of $H$-stable subspaces induces homotopy equivalence, then in fact $f$ admits an equivariant homotopy-inverse.

You can find all this and much more in Peter May's lovely notes here.

One set of sufficient conditions may be obtained if your map plays nicely with respect to subspaces fixed by closed subgroups. Let $X$ and $Y$ be $G$-spaces for any $G$ (not only the torus) and assume that you have an equivariant map $f:X \to Y$. If for any closed subgroup $H < G$ the induced map $X^H \to Y^H$ of $H$-stable subspaces induces homotopy equivalence, then in fact $f$ admits an equivariant homotopy-inverse.

You can find all this and much more in John Greenlees and Peter May's lovely notes here.