If the sun suddenly stopped emitting neutrinos, what might we infer (after checking that our neutrino detectors were still operational) ?

The best inference would be that Fusion reactions in the Sun have ceased within the past few minutes. This is because neutrinos are created from nuclear decay at the core of nuclear reactors such as that at the core of the sun. The sun is estimated to produce approximately 65 billion (6.5*1010) neutrinos per second per square centimeter.