Subir Sarkar proposed that dark energy does not exist, i.e. the cosmological constant of our universe may not be positive (https://arxiv.org/abs/2106.03119). His stance was based on the 740 type Ia supernovae in the above linked Arxiv paper. Other than the data from these 740 type Ia supernovae, are there any other evidence that shows that dark energy does not exist?
As you can see from charts like these (left column), CMB and BAO data together suggest $Ω_Λ\approx 0.7$ without any dependence on the supernova data.
I think Sarkar et al's paper is just wrong. They're suggesting that the standard treatment of the effect of our peculiar velocity on our observations may not be correct. That's impossible, because it depends only on the physics of the interaction of light with our instruments in the here-and-now, which is extremely well understood (because it can be investigated with actual repeatable experiments, and not just passive cosmological observations). It's independent of the origin of the light, and therefore of the cosmological model.
They base their analysis on "The deceleration parameter in 'tilted' Friedmann universes" by Tsagas and Kadiltzoglou (DOI (paywalled), arXiv), which seems to be about test particles with peculiar velocities in exact FLRW spacetimes (not perturbed spacetimes where the actual matter has a peculiar velocity).
The authors of both papers seem to suffer from the common-ish misconception that to analyse someone's observations correctly, you have to use "their" coordinate system. The truth is just the opposite: all coordinate systems are equivalent, so you can use whichever one makes your problem easiest. For FLRW cosmology, FLRW coordinates make almost every problem easier, and so cosmologists use them. Tsagas and Kadiltzoglou define analogues in their tilted coordinates of values like the deceleration parameter of the title, and suggest that those are what we really measure, but they aren't.
To make the discussion simpler, testing the model simply means extracting some constraint from the theory and then confirming it experimentally. For FLRW sum rule it means we have to measure each $\Omega$ component separately, add them up and see if the sum equals 1. The cosmological constant as derived from the sum rule is a prediction made by using the model. The model will be tested when we measure late time ISW. Hence, the current Hubble tension is a tension between a universe-model-free observation of accelerating supernovae (which does not have to be monopolar like due to $\Lambda$. There exist $>3\sigma$ fits assuming dipolar acceleration due to bulk flow) and a universe-model-depdendent prediction of a free parameter from BAO+CMB etc.
CMB and BAO prove dark energy only if we assume that FLRW holds. The question is what we are doing here? Are we using the model or are we testing it? Dark energy is only a conclusion of the cosmic sum rule $(\Omega_m+\Omega_k + \Omega_\gamma +\Omega_\Lambda =1)$. If you claim that FLRW works fine from now till almost the beginning of the universe, then you can measure all but the last term, correct for the redshift and finally solve for $\Omega_\Lambda$. However, if you want to test it, you have to measure all these components for each bin of redshift separately, i.e. you need to confirm this rule holds at z=0.1, z=1, z=10 ... z=1100 separately. But this is not what is done here (or anywhere for that matter). Until we measure late-time ISW we cannot really say much about the measurement of $\Lambda$.
In this paper no model is assumed. Instead they simply analyse semi-local behaviour. So when they find a local dipolar Hubble constant, nonconvergence of local flow beyond 100Mpc (even beyond 200Mpc), differing velocity requirements to account for CMB dipole and flux Doppler effect for radio sources, strange alignment of CMB quadrupole and octupoles, one has to explain why locally FLRW with perturbations is simply not enough. But really we need to wait for more data. For $5\sigma$ we usually need several millions of sources to perform maximum likelihood analysis, i.e. tell the signal from the noise. Currently, very often you only see $\chi^2$ which is simply fitting a model we assume to hold.
There is now strong observational evidence against the assumption of large-scale homogeneity & isotropy (i.e. FLRW metric) upon which analyses of supernovae, CMB, BAO etc rely, in order to infer that Universe is dominated by dark energy. Secrest et al. (Astrophys. J. Lett. 937 (2022) L31) demonstrate with a significance exceeding 5σ that the distribution of distant matter as traced by radio sources & quasars is not isotropic in the reference frame in which the CMB is isotropic. As MadMax and CosmoJ have said, the inference of dark energy is then suspect too as it relies on the validity of the cosmic sum rule (which is just a restatement of the Friedmann-Lemaitre equation for FLRW). It would appear that the real Universe disagrees with the assertion by benrg that "all coordinate systems are equivalent"!