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We study the effects of dilute La and Rh substitutional doping on the electronic structure of the relativistic Mott insulator Sr2IrO4 using fully relativistic and magnetically noncollinear density functional theory with the inclusion of an on-site Hubbard U. To model doping effects, we have adopted the supercell approach, that allows for a realistic treatment of structural relaxations and electronic effects beyond a purely rigid band approach. By means of the band unfolding technique we have computed the spectral function and constructed the effective band structure and Fermi surface (FS) in the primitive cell, which are readily comparable with available experimental data. Our calculations clearly indicate that La and Rh doping can be interpreted as effective electron and (fractional) hole doping, respectively. We found that both electron and hole doping induce an insulating-to-metal transition (IMT) but with different characteristics. In Sr2-xLaxIrO4 the IMT is accompanied by a moderate renormalization of the electronic correlation substantiated by a reduction of the effective on-site Coulomb repulsion U-J from 1.6 eV (x=0) to 1.4 eV (metallic regime of x=12.5%). The progressive closing of the relativistic Mott gap leads to the emergence of connected elliptical electron pockets at (π/2,π/2) and less intense features at X on the Fermi surface. The average ordered magnetic moment is slightly reduced upon doping, but the canted antiferromagnetic state is perturbed on the Ir-O planes located near the La atoms. The substitution of Ir with the nominally isovalent Rh is accompanied by a substantial hole transfer from the Rh site to the nearest-neighbor Ir sites. This shifts down the chemical potential, creates almost circular disconnected hole pockets in the FS, and establishes the emergence of a two-dimensional metallic state formed by conducting Rh planes intercalated by insulating Ir planes. Finally, our data indicate that hole doping causes a flipping of the in-plane net ferromagnetic moment on the Rh plane and induces a magnetic transition from the antiferromagnetic (AF)-I to the AF-II ordering. [ABSTRACT FROM AUTHOR]