Fermi liquid theory is remarkably successful in describing the transport and optical properties of metals; at frequencies higher than the scattering rate, the optical conductivity adopts the well-known power law behavior σ1(ω) ∝ ω(-2). We have observed an unusual non-Fermi liquid response σ1(ω) ∝ ω(-1±0.2) in the ground states of several cuprate and iron-based materials which undergo electronic or magnetic phase transitions resulting in dramatically reduced or nodal Fermi surfaces. The identification of an inverse (or fractional) power-law behavior in the residual optical conductivity now permits the removal of this contribution, revealing the direct transitions across the gap and allowing the nature of the electron-boson coupling to be probed. The non-Fermi liquid behavior in these systems may be the result of a common Fermi surface topology of Dirac cone-like features in the electronic dispersion.