genetics - Are there dog breeds that are so far apart genetically that they can't produce viable offspring?

I actually found some sort of reference for this. Apparently in the case of a beagle and irish setter pairing, they had a lot of difficulty in producing pups, but as breeds have a lot of genetic quirks, this might be due to a genetic accident; this is probably a case of mutual infertility rather than speciation. There are probably mutations in their genomes which are causing non viability in the pups the way that it does with some human parents who have difficulty having children - an immunological incompatibility or traits which cause most offspring non-viable.

In fact the discussion points out that the species designation does not always mean they cannot produce offspring, but simply do not. Coyotes and wolves will produce offspring too if they are encouraged to do so, but are competing and antagonistic in the wild and so never do mate in practice.

So the answer is probably no. (the beagle/setter pups did show up, but with a much smaller litter than usual with just 2 pups).