Surjectivity of a continuous map implies surjectivity on $\pi_1$

No for general spaces.

Following Qiaochu Yuan:

Let $X$ be the Warsaw circle (the version that is path-connected, like this). Let $Y=S^1$. Radially project $X$ to $Y$. The projection cannot induce a surjective homomorphism between the fundamental groups, as $X$ is simply connected. The fibers of the projection maps over every point is a singleton except for one exceptional point where the fiber is a line.

Edit:

The main theorem of this paper of Smale seems to answer your question. (I think $LC^n$) means that it is locally $n$-connected, but Smale doesn't seem to define this.

One good way to rule out pathologies such as the Warsaw circle is to restrict ourselves to the piecewise linear category. I think that in this case, the map is always onto in $\pi_1$.

Suppose that $\gamma: S^1 \to Y$ is a loop in $Y$. Since everything is PL, look at the original map $f$ as a map from the $1$-skeleton of $X$ to the $1$-skeleton of $Y$, a homomorphism of undirected graphs. $\gamma$ is now a finite loop of of directed edges in $Y$. Because $f$ is onto, each of these edges can be lifted to $X$. Say we have two consecutive edges $y$ and $y'$, where $y = (y_0, y_1)$, $y' = (y'_0, y'_1)$, and $y_1 = y'_0$. These lift to $(x_0, x_1)$ and $(x'_0, x'_1)$.

But also, the fiber $f^{-1}(y_1)$ is connected, so there is a path in that fiber from $x_1$ to $x'_0$. So we can stitch together the lifted edges to get a finite loop $\beta$ in $X$ which is a lift of $\gamma$ up to homotopy: $f \circ \beta \sim \gamma$. Now at the level of $\pi_1$, $f^*([\beta]) = [\gamma]$, so $f^*$ is onto.