We present new results on the iron dynamics in the icosahedral quasicrystal i-AlCuFe and two cubic approximants as well as the non-approximant Al-Cu-Fe cubic B2 phase. Conventional Mössbauer spectroscopy is used as well as, for the i-AlCuFe phase, high Doppler velocity Mössbauer spectroscopy and quasielastic neutron scattering for samples with different Fe isotope contents. We show that in the i-phase the Fe Lamb-Mössbauer recoilless fraction decreases below that predicted for lattice vibrations alone for temperatures above about 550 K. This decrease is correlated with the onset of a quasielastic signal seen in both Mössbauer and neutron backscattering spectroscopy, which indicates the presence above 550 K of Fe jump processes confined in a local cage. The timescale of the Fe jumps (660 ps at 1000 K) and their temperature dependence differ widely from those of Cu jumps in the same i-AlCuFe quasicrystal. From the temperature dependence of the quadrupole splitting of the (57)Fe Mössbauer spectrum, one can distinguish two kinds of Fe jumps, one starting at 550 K and the second above 800 K. In the two cubic approximants, a loss in the Fe recoilless fraction also occurs above 550 K, revealing the same kind of Fe dynamics as in the i-phase but the effect is smaller. On the other hand, no anomalous Fe dynamics (other than lattice vibrations) is detected in the B2-AlCuFe phase. Since the cubic approximants possess similar local configurations as the quasicrystal, we conclude that locally a Penrose tile description is appropriate. This shows that the detected Fe jumps can be interpreted in terms of phason-like local tiling flips.