Icosahedral Al-Cu-Fe quasicrystal (QC) shows moderate electrical conductivity and low thermal conductivity, and both p- and n-type conduction can be controlled by tuning the sample composition, making it potentially suited for thermoelectric materials. In this work, we investigated the effect of introducing chemical disorder through heavy element substitution on the thermal conductivity of Al-Cu-Fe QC. We substituted Au and Pt elements for Cu up to 3 at% in a composition of Al63Cu25Fe12, i.e., Al63Cu25−x(Au,Pt)xFe12 (x = 0, 1, 2, 3). The substitutions of Au and Pt for Cu reduced the phonon thermal conductivity at 300 K (κph,300K) by up to 17%. The reduction of κph,300K is attributed to a decrease in the specific heat and phonon relaxation time through heavy element substitution. We found that increasing the Pt content reduced the specific heat at high temperatures, which may be caused by the locked state of phasons. The observed glass-like low values of κph,300K (0.9–1.1 W m−1 K−1 at 300 K) for Al63Cu25−x(Au,Pt)xFe12 are close to the lower limit calculated using the Cahill model.