The influence of hydrogen presence on the stability of iron phases (bcc, hcp, dhcp, fcc, simple cubic) in a wide pressure interval at 0 K has been studied by the first-principles projector augmented-wave (PAW) method. Hydrogen is shown to occupy different interstitial lattice positions depending on the type of structure and pressure. An introduction of hydrogen impurities (~6 at. %) leads to a stabilization of the close-packed iron structures, shifting the calculated pressure of the bcc-hcp transition from ~9 GPa for pure iron to 7 GPa for Fe (6 at. % H). This tendency is further enhanced in the iron hydride structures. The iron hydrides in the close-packed structures (hcp, dhcp, fcc) are essentially degenerate in energy and found to be most stable in the whole pressure range.