Luminous red novae (LRNe) are transients characterized by low luminosities and expansion velocities, and they are associated with mergers or common-envelope ejections in stellar binaries. Intermediate-luminosity red transients (ILRTs) are an observationally similar class with unknown origins, but they are generally believed to be either electron-capture supernovae in super-asymptotic giant branch stars or outbursts in dusty luminous blue variables (LBVs). In this paper, we present a systematic sample of eight LRNe and eight ILRTs detected as part of the Census of the Local Universe (CLU) experiment on the Zwicky Transient Facility (ZTF). The CLU experiment spectroscopically classifies ZTF transients associated with nearby (<150 Mpc) galaxies, achieving 80% completeness for m r < 20 mag. Using the ZTF-CLU sample, we derive the first systematic LRNe volumetric rate of ${7.8}_{-3.7}^{+6.5}\times {10}^{-5}$ Mpc-3 yr-1 in the luminosity range -16 ≤ M r ≤ -11 mag. We find that, in this luminosity range, the LRN rate scales as ${dN}/{dL}\propto {L}^{-2.5\pm 0.3}$ -significantly steeper than the previously derived scaling of L -1.4±0.3 for lower-luminosity LRNe (M V ≥ -10 mag). The steeper power law for LRNe at high luminosities is consistent with the massive merger rates predicted by binary population synthesis models. We find that the rates of the brightest LRNe (M r ≤ -13 mag) are consistent with a significant fraction of them being progenitors of double compact objects that merge within a Hubble time. For ILRTs, we derive a volumetric rate of ${2.6}_{-1.4}^{+1.8}\times {10}^{-6}$ Mpc-3 yr-1 for M r ≤ -13.5 mag, which scales as ${dN}/{dL}\propto {L}^{-2.5\pm 0.5}$ . This rate is ~1%-5% of the local core-collapse supernova rate and is consistent with theoretical ECSN rate estimates.