Andrew's answer is right, but I'll just throw in a few comments since "topological" homotopy invariants are of great interest to me. Here Paul Fabel has shown that $\pi_{1}^{top}$ on the Hawaiian earring is not a topological group. It turns out that multiplication can fail to be continuous even for some reasonably nice spaces (like locally simply connected planar continua). This and a nice connection with free topological groups appears in a manuscript I just posted on arXiv (perhaps shameless self-promotion...but I'll post the link when it becomes available).
This quotient map mistake has appeared many places in the literature, even in an appendix by Peter May from the 70's who described the topological fundamental groupoid (giving each hom-set the quotient topology of fixed endpoint Moore path spaces). The same false assertion that products of quotient maps are again quotient maps was also used to "show" the higher topological homotopy groups are topological groups.
With this quotient topology, the topological fundamental group(oid) is discrete on spaces that have universal covers (are path connected, locally path connected, and semi-locally simple connected). When it is non-discrete, this topology is often difficult to deal with. After all...there are going to be many homotopic loops that we identify in the quotient but which "look nothing alike" in the space.
While we don't always have a topological group, we're not completely out of luck. $\pi_{1}^{top}(X)$ is always a quasitopological group in the following sense:
definition: A quasitopological group is a group $G$ with topology such that inversion is continuous and multiplication $G\times G\rightarrow G$ is continuous in each variable.
A basic theory of these objects can be found in "Topological Groups and Related Structures" some of which you can get on google books.
Edit: The topological fundamental group and free topological groups is the new paper I mentioned. It is a bit denser than the one from Andrew's answer but if anyone is interested enough to read it, I'd greatly appreciate any comments, suggestions, or corrections.