  Luminous red novae (LRNe) are transients characterized by low luminosities
and expansion velocities, and 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 generally
believed to either be electron capture supernovae (ECSN) in super-AGB stars, or
outbursts in dusty luminous blue variables (LBVs). In this paper, we present a
systematic sample of 8 LRNe and 8 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$^{+6.5}_{-3.7}\times10^{-5}$ Mpc$^{-3}$ yr$^{-1}$ in the luminosity range
$-16\leq$M$_{\rm{r}}$$\leq -11$ mag. We find that in this luminosity range, the
LRN rate scales as dN/dL $\propto L^{-2.5\pm0.3}$ - significantly steeper than
the previously derived scaling of $L^{-1.4\pm0.3}$ for lower luminosity LRNe
(M$_{V}\geq-10$). 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$_{r}\leq-13$
mag) are consistent with a significant fraction of them being progenitors of
double compact objects (DCOs) that merge within a Hubble time. For ILRTs, we
derive a volumetric rate of $2.6^{+1.8}_{-1.4}\times10^{-6}$
Mpc$^{-3}$yr$^{-1}$ for M$_{\rm{r}}\leq-13.5$, that scales as dN/dL $\propto
L^{-2.5\pm0.5}$. This rate is $\approx1-5\%$ of the local core-collapse
supernova rate, and is consistent with theoretical ECSN rate estimates.
