The correct way to write this is:
Do you think you are smarter than me?
"Than" is a conjunction or preposition, whereas "then" is an adverb, noun, or adjective. I do not believe there is a situation where they are interchangeable.
A brief explanation which illustrates the differences well can be found here.
(I originally suggested that it should be written "Do you think you are smarter than I?" I rescinded it because @Kosmonaut pointed out that many people use "me" instead. His comment links to 2 discussions on the subject. I don't believe that "I" is incorrect, but most people probably wouldn't find fault with your using "me" instead, and I don't want to derail this thread any more.)