In the framework of a systematic study with the ALMA interferometer of IR-selected main-sequence and starburst galaxies at z similar to 1-1.7 at typical similar to 1 '' resolution, we report on the effects of mid-IR- and X-ray-detected active galactic nuclei (AGN) on the reservoirs and excitation of molecular gas in a sample of 55 objects. We find widespread detectable nuclear activity in similar to 30% of the sample. The presence of dusty tori influences the IR spectral energy distribution of galaxies, as highlighted by the strong correlation among the AGN contribution to the total IR luminosity budget (f(AGN) = L-IR,L-AGN/L-IR), its hard X-ray emission, and the Rayleigh-Jeans to mid-IR (S-1.2 mm/S-24 mu m) observed color, with evident consequences on the ensuing empirical star formation rate estimates. Nevertheless, we find only marginal effects of the presence and strength of AGN on the carbon monoxide CO (J = 2, 4, 5, 7) or neutral carbon ([CI](P-3(1) - P-3(0)), [CI](P-3(2) - P-3(1))) line luminosities and on the derived molecular gas excitation as gauged by line ratios and the full spectral line energy distributions. The [CI] and CO emission up to J = 5, 7 thus primarily traces the properties of the host in typical IR luminous galaxies. However, our analysis highlights the existence of a large variety of line luminosities and ratios despite the homogeneous selection. In particular, we find a sparse group of AGN-dominated sources with the highest L-IR,L-AGN/L-IR,L- SFR ratios, greater than or similar to 3, that are more luminous in CO (5-4) than what is predicted by the L'(CO(5-4)) -L-IR,L- SFR relation, which might be the result of the nuclear activity. For the general population, our findings translate into AGN having minimal effects on quantities such as gas and dust fractions and star formation efficiencies. If anything, we find hints of a marginal tendency of AGN hosts to be compact at far-IR wavelengths and to display 1.8 times larger dust optical depths. In general, this is consistent with a marginal impact of the nuclear activity on the gas reservoirs and star formation in average star-forming AGN hosts with L-IR > 5 x 10(11) L-circle dot, typically underrepresented in surveys of quasars and submillimeter galaxies.