We investigate with angle-resolved photoelectron spectroscopy the changes of the Fermi surface and the main bands from the paramagnetic state to the antiferromagnetic (AFM) state occurring below 72 K in Fe1.06Te. The evolution is completely different from that observed in Fe pnictides, as nesting is absent. The AFM state is a rather good metal, in agreement with our magnetic band structure calculation. On the other hand, the paramagnetic state is very anomalous with a large pseudogap of ~65 meV on the electron pocket that closes in the AFM state. We discuss this behavior in connection with spin fluctuations existing above the magnetic transition and the correlations predicted in the spin-freezing regime of the incoherent metallic state.