We report on IRAM 30-m/EMIR observations of 38 Herschel sources chosen as the brightest red submillimeter (submm) sources in 18 Planck-selected fields drawn from the Planck high-z (PHz) sample of protocluster candidates. These fields host overdensities of red Herschel sources, with high star formation rates (∼10 000 M_⊙ yr⁻¹), as obtained from the Planck measurements. The goals of these observations are to measure the spectroscopic redshifts of the Herschel sources in the PHz fields, investigate the origin of their bright submm emission, and find evidence of their association with high-z protoclusters. We detected 40 CO lines on a total of 24 bright (S​_{350 μm} > 40 mJy) Herschel sources in 14 of the 18 PHz fields. The measured average redshift is ⟨z_CO⟩ = 2.25 ± 0.09, spanning a range from 1.32 to 2.75. We measured the redshifts for multiple Herschel sources in projected proximity in eight PHz fields. In half of those fields, we detected between two and three objects at similar redshifts, supporting the idea that the PHz fields contain high-z protoclusters. The detection of sources at different redshifts in the same field demonstrates that foreground and background sources also contribute to the total submm emission. We compared the properties of the molecular gas and of the star formation activity of our sources with samples of normal star-forming galaxies (SFGs), submm galaxies (SMGs), and CO-detected cluster and protocluster galaxies drawn from the literature at similar redshifts. We find that the PHz-IRAM sources are mainly normal SFGs, with only ∼20% undergoing a starburst phase. The PHz-IRAM sources are characterized by star formation rates (⟨SFR⟩ = 1043 ± 157 M_⊙ yr⁻¹) and gas masses (⟨M_gas⟩ = (4.0 ± 0.7) × 10¹¹ M_⊙) that are, on average, eight and five times higher than those typical of normal SFGs at similar redshifts. Their dust temperatures (⟨T_dust⟩ = 29.2 ± 0.9 K) and depletion timescales (⟨τ_dep⟩ = 0.47 ± 0.07 Gyr) are instead consistent with those of normal SFGs. The analysis of the CO spectral line energy distribution, available for ten PHz-IRAM sources, peaks at a low quantum rotation number (J_up = 3) in most of the cases, implying low gas excitation. These properties imply that a significant number of PHz-IRAM sources contain extended and cold molecular gas reservoirs at low excitation and that their star formation is driven by secular processes. Multiplicity and moderate gravitational lensing might also play a role in producing the observed properties. Some of these properties are also observed in other CO-detected infrared-luminous protocluster galaxies at z ≃ 1.3 − 3. We find that the protoclusters with the highest level of star formation, drawn from the largest volume simulations available to date, exhibit similar SFRs as compared to the PHz protoclusters; however, they separate out into a higher number of star-forming galaxies. Millimeter and CO observations at higher spatial resolution than those presented here would be necessary to further elucidate the properties of our PHz-IRAM sources and determine which mechanisms drive star formation in infrared-luminous protocluster galaxies.