At this point, even if most of the species of dog that people own went extinct, I doubt the consequences would actually be that sever. Which is mainly because "house dogs" don't really keep a balance in nature these days.
Don't get me wrong; I like dogs. It's just that the "house dogs" aren't really predators or prey these days. Of course, if you're a person who calls animals like wolves and such dogs, then it could bring serious consequences.