Is FeS2 both an ionic and covalent compound?
Why do you feel a need to classify it under one or another? Chemistry is not about having a list of ionic compounds and another list of covalent compounds.

There are "ionic" bonds (see below) between $\ce{Fe^2+}$ and $\ce{S2^2-}$, and there is a $\ce{S-S}$ covalent bond in $\ce{S2^2-}$. That's all there is to it.

If I absolutely had to pick one, I would say it is an ionic compound. $\ce{S2^2-}$ is a covalently bonded species; $\ce{FeS2}$ is an ionic crystalline solid. The fact that it melts at $1193~^\circ\mathrm{C}$ supports this "classification".



As @Mith notes in the comments, all ionic bonds are not 100% ionic, and a lot of ionic bonds are in fact covalent to a large degree. The fact that $\ce{Fe^2+}$ is a transition metal ion (meaning it has empty 3d and 4s orbitals available for overlap) and also that sulfur is a rather polarisable ion (because of its larger size) means that there is likely to be significant covalency in the $\ce{Fe-S}$ bonds.

What does this mean? It just shows that trying to classify something as "ionic" or "covalent" is futile. $\ce{FeS2}$ is likely somewhere right in the middle of the spectrum.