This paper presents the results of a detailed theoretical investigation of the iron line formation NLTE problem in a three-dimensional model of the solar photosphere, which we have obtained from a very recent radiation hydrodynamics simulation of solar surface convection. In this first paper we have neglected the effects of horizontal radiative transfer on the atomic level populations, but we have considered a realistic atomic model for iron that contains hundreds of radiative transitions from the UV to the IR. The self-consistent solutions of the kinetic and transfer equations have been obtained with a new NLTE code, which is based on very efficient iterative methods. We find that overionization due to the near-UV radiation field does take place but mainly in the granular atmospheric regions. This well-known NLTE mechanism tends to produce underpopulation of all the Fe I levels and a very small overexcitation of the Fe II levels. All over the three-dimensional photospheric model Fe II is the dominant ionization stage. We find significant LTE versus NLTE discrepancies mainly for the low-excitation Fe I lines. This applies to both the vertically emergent profiles from the granular regions and also to the spatially averaged profiles. These discrepancies are due to the line opacity deficits that result from the aforementioned underpopulation of the Fe I levels. The emergent profiles of the low-excitation lines of Fe I are thus weaker in NLTE than in LTE. In particular, the largest errors in the equivalent widths (due to the LTE assumption) are found for the weakest low-excitation lines of Fe I. We also give quantitative estimates of the errors in the temperature structure of semiempirical solar granulation models obtained via the application of LTE inversion techniques to several groups of Fe I lines. For instance, the widely used Fe I 6301 and 6302 Å lines tend to lead to an overestimation of about 100-200 K in the granular regions but to a similar underestimation in the intergranular plasma. The present paper considers also the case of the Sun observed with low spatial resolution, with particular emphasis on the long-standing iron abundance problem. We show that it is possible to obtain a very good fit to the observed spectral line shapes by slightly changing the iron abundance (for both the LTE and NLTE cases). In general, the iron abundance we need for reaching the best NLTE fit to observed equivalent widths is 0.074+/-0.03 dex larger than that needed to obtain the best LTE fit. Our most relevant conclusion with regard to the solar iron abundance issue is the following: if NLTE effects are fully taken into account in the three-dimensional model of the solar photosphere, we obtain the meteoritic iron abundance value (AFe=7.50). However, if the abundance analysis is done assuming LTE, we find AFe=7.43, in close agreement with the recent LTE analysis of Asplund and collaborators. Our results do indicate that NLTE effects are significant but not above the 0.1 dex level in the Sun. We consider our NLTE result for the iron abundance as an additional hint of the realism of such three-dimensional hydrodynamic simulations. We conclude that the success of the LTE fitting approach is no proof that NLTE effects are negligible because the existing NLTE effects are compensated for in the LTE analysis by a change in the derived iron abundance. The paper ends emphasizing the great importance of a full three-dimensional NLTE approach in order to be able to lead to new advances in the field of quantitative stellar spectroscopy and, in particular, for a correct derivation of elemental abundance ratios in the atmospheres of metal-poor stars.