This follows from a 1931 theorem of M. H. A. Newman. Theorem 1 of the paper says that for a manifold $M$ with a metric and an integer $p>1$, there is a constant $d$ so that any (uniformly continuous) periodic transformation of $M$ of order $p$ must move a point of $M$ at least a distance $d$.
Given a homeomorphism $f$ of a torus $T^n$ which acts trivially on $\pi_1(T^n)$ and fixes a point, there is a lift of $f$ to any cover of $T^n$ which has a fixed point. Moreover, $f$ is homotopic to the identity, since $T^n$ is a $K(\mathbb{Z}^n,1)$. Then in some Euclidean metric on $T^n$, every point will be homotopic to its image under $f$ by a path of length at most $D$ (the tracks of the homotopy have bounded length). Pass to the $m^n$-fold cover of $T^n$ (coming from $ker\{\mathbb{Z}^n\to (\mathbb{Z}/m\mathbb{Z})^n\}$), and rescale the metric on this cover by $1/m$, every point in this cover will be distance at most $D/m$ from its image under the lift of $f$ to this cover with a fixed point, which also has period $p$ if $f$ does. Taking $m$ large enough (so that $D/m <d$ from Newman's theorem), Newman's theorem implies that the lift of $f$ to this cover will be the identity if $f$ has period $p$, and hence $f$ is the identity.
Hence for a finite group $G$ acting (faithfully) on a torus $T^n$, every non-trivial element $f$ of $G$ must act non-trivially on $\pi_1(T^n)$.