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On a plot of $\Omega_\Lambda$ versus $\Omega_M$, there are three sets of observations that provide constraints: supernovae, the cosmic microwave background, and baryon acoustic oscillations. These three regions in the $\Omega_\Lambda$-$\Omega_M$ plane all have a common region of intersection, which is quite small. If they had failed to overlap, it would have proved that there was a lack of consistency in cosmological models. If future improvements in the data reduce the sizes of these regions and they then fail to overlap, it will be the same problem.
At one time there were claims that the oldest globular clusters were older than the age of the universe inferred from Hubble expansion. If this had been correct, then it would have been evidence against Big Bang cosmology, or at least against a particular model. But in fact we're now living in the age of high-precision cosmology, and current data show that the clusters are younger than the universe after all.
Big Bang nucleosynthesis makes specific predictions about the relative abundances of various isotopes. In most cases, these predictions are correct, but there still seems to be a problem with the 7Li/H ratio. I don't think anyone is about to throw away all of modern cosmology over this issue, but at some point it needs to be resolved, or there's something wrong with either our cosmological models or our knowledge of nuclear physics.
It's possible according to GR for the universe to be rotating. Solar system observations allow us to put an upper limit on the rate of rotation. If the rate of rotation were proved to be nonzero, then it would violate the assumption of isotropy in Friedmann models. However, the observational constraints are already tight enough that they don't allow rotation to be a huge effect (e.g., centrifugal forces can't contribute significantly to cosmological expansion), so Friedmann models would still probably be good at least as approximations.
General relativity is not a self-consistent theory if energy-momentum is not locally conserved. Therefore any evidence of local nonconservation of energy-momentum would indicate that GR needed to be revised, and thus our cosmological models would also probably need to be revised. There is a history of people like Fred Hoyle seriously proposing that matter was spontaneously created in a vacuum (by something he called a "C field"). This hypothesis is testable, and there is currently no believable evidence for it.
How do you disprove that redshift is determined by stretching of space, how do you disprove that space can expand and that just a metric devised by a theorist can make it expand?
There have been serious proposals by Hoyle's supporters that redshifts were not purely cosmological but were at least partly "intrinsic." This would require some kind of unspecified modification to standard quantum mechanics, and there is no evidence to support such a modification. As explained in this article by Ned Wright, attempts to construct cosmological models using these ideas have not been successful; the resulting models are inconsistent with observation.