Does iron(VI) fluoride exist? Why does $\ce{FeF6}$ not exist? There are hexavalent iron compounds, so that is not the problem. There is the $\ce{[Fe(CN)6]^3-}$ ion, so there doesn't seem to be a steric problem.
Besides steric factors related to the small size of the transition metal core, we could be seeing an electronic effect described in this answer.  Iron is fairly early in the transition metal series, so when pushed beyond the $+3$ oxidation state it has few $d$ electrons in the central core.  As explained in the referenced answer, this makes the iron strongly pi-electron accepting, and therefore more favorable for combining with a stronger pi-donor ligand such as oxide instead of fluoride.