No, I know of no Earth Scientist who would contest the idea that continents are moving (along with the larger plates that they are part of). That doesn't mean that some uninformed people think continental drift is a hoax, but then again, some people mistakenly believe that Earth is flat. Sometimes even the strongest scientific ideas fail to convince everyone.

No, continental drift is such a thoroughly supported theory that nobody seriously doubts it. There are still some unanswered questions about exactly what causes it and how it happens, though.