Do viruses suffer from Muller's ratchet? Can it significantly influence the current coronavirus pandemic?
As with most things in the natural world, there isn't really a single straightforward answer to this question! It depends largely on the ecological contect and the population dynamics.
Basically, in organisms with high mutations rates (such as RNA viruses), Muller's Ratchet suggests that the mean fitness in a population will always decrease. In small populations, mutation-free individuals are rare, and they will be lost through genetic drift. In 1990, Chao showed that deleterious mutations are generated at a sufficiently high rate to advance Muller's ratchet in an RNA virus.
However, in general, Muller's ratchet can be argued to be ineffective in viral populations because populations are never 'small', and drift is usually very weak (or selection is strong, in other words). Chao also notes that the effective population size in nature, 8x10^9 is often irrelevant to whether or not Muller's Ratchet is active, because Viruses often pass through the bottleneck of a single individual. Additionally, frequently recombining viruses such as HIV can also be subject to Muller's Ratchet Yuste et al (1999). The results of Chao have been generalised to many other viruses.
However, these studies are lab-based and don't pertain to your question about what might happen to SARS-CoV-2 'in the wild'. RNA viruses show the highest mutations rates in nature (Sanjuan et al 2010). This can lead to Muller's Ratchet in small populations where selection is weak and drift is prominent, but can lead to high levels of adaptation in large populations. It is not just the census population size which is important, but the effective population size, which can be thought of as a proxy for the amount of genetic diversity within a population. If a population rapidly expands from a single origin, as is expected to have happened with the SARS-CoV-2 outbreak, then the census population size would be larger than the expected diversity. Novella (1995/1996) showed that there is an interplay with the size of the bottleneck and the fitness of the surviving clones; when a more fit viruses passes through the bottleneck, they are more likely to avoid Muller's Ratchet. So in the case of SARS-CoV-2, whether or not the virus would be subject to Muller's Ratchet would depend on both the variation and fitness of mutation of the viruses who jumped from the intermediate animal to humans.
At the risk of overcomplicating things, population/deme structure also plays a role. If the virus evolves in different demes within a structure population, viruses within a deme can evolve population specific mutations. If there is then gene-flow between the demes, there can exist selection between the different mutations, driving up the average fitness across demes (Mirales et al 2009) and avoiding the Ratchet.
Muller's ratchet is more relevant/more important in organisms with small effective population sizes. While the effective population size of the worldwide SARS-COV-2 population is smaller than (billions of copies per person [a typical detection threshold is 10^4 copies/milliliter of blood]) x (millions of infected people), it's very likely to be so large that Muller's ratchet is irrelevant.
Other answers cover well Muller's ratchet of the worldwide virus population.
However, drug-induced Muller's ratchet (aka mutational meltdown) might be considered on the level of the viral population that infected one patient. A recent publication Considering mutational meltdown as a potential SARS-CoV-2 treatment strategy talks about potentially inducing mutational meltdown on the within-patient viral population.
Note it's not research but a comment paper! Authors review the theory and empirical evidence that makes them think that mutational meltdown has the potential to become a coronavirus treatment.