Does any iron fuse in stars before they go supernova?

The "iron core" in a supernova is actually the end product of a nuclear statistical equilibrium that begins when the silicon core begins to fuse with alpha particles (helium nuclei). Exothermic reactions are possible right up to Nickel-62 (which is actually the nucleus with the highest binding energy per nucleon). In fact, successive, rapid alpha captures produce nuclei with the same number of protons and neutrons, but at the same time, the competing processes of photodisintegration and radioactive decay work in the other direction. The process is thought to mostly stop at Nickel-56 which, because heavier nuclei are more stable with $n/p>1$, then undergoes a couple of $beta^{+}$ decays via Cobalt-56 to Iron-56. However, the core of a supernova just before it explodes is likely to contain a bit of a mixture of iron-peak isotopes.

Before all this happens it is possible for iron and nickel to undergo nuclear reactions if there is an appropriate source of free neutrons. The elements beyond iron in our universe are predominantly created by neutron-capture in either the r-process or the s-process.

The r-process is thought to occur after a core-collapse supernova (or a type Ia supernova) has been initiated. The neutron flux is created by the neutronisation of protons by a dense, degenerate electron gas in the collapsing core.

However, the s-process can occur outside the core of a massive star before it explodes. It is a secondary process because it needs iron nuclei to be present already - that is, the iron that is used for the seed nuclei is not produced inside the star, it was already present in the gas from which the star formed. The s-process in massive stars uses free neutrons produced during neon burning (so at advanced nuclear burning stages beyond helium, carbon and oxygen burning) and results in the addition of neutrons to iron nuclei. This builds up heavy isotopes, which may either be stable or undergo $beta$ decay and/or further neutron captures to build up a chain of "s-process elements" (e.g. Sr, Y, Ba) all the way up to lead. The overall process is endothermic, but the yields and reaction rates are so small that it has no major influence on the overall energetics of the star. The newly-minted s-process elements are easily blasted into the interstellar medium shortly afterwards when the supernova explodes.