In the framework of a systematic ALMA study of IR-selected main-sequence and starburst galaxies at z~1-1.7 at typical ~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 detectable nuclear activity in ~30% of the sample. The presence of dusty tori influences the IR SED of galaxies, as highlighted by the strong correlation among the AGN contribution to the total IR luminosity budget (fAGN = LIR,AGN/LIR), its hard X-ray emission, and the Rayleigh-Jeans to mid-IR (S1.2mm/S24um) observed color, with consequences on the empirical SFR estimates. Nevertheless, we find only marginal effects of AGN on the CO (J=2,4,5,7) or neutral carbon ([CI](1-0), [CI](2-1)) line luminosities and on the derived molecular gas excitation as gauged by line ratios and the full SLEDs. The [CI] and CO emission up to J=5,7 thus primarily traces the properties of the host in typical IR luminous galaxies. However, we highlight 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 LIR,AGN/LIR,SFR ratios, >3, that are more luminous in CO(5-4) than what is predicted by the L'CO(5-4)-LIR,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 SFEs. If anything, we find hints of a marginal tendency of AGN hosts to be compact at far-IR wavelengths and to display 1.8x 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 LIR>5e11 Lsun, typically underrepresented in surveys of quasars and SMGs.