I would not say that there are grammatical structures which can only be used in writing. However, there are definitely stylistic differences between a dialect (spoken version) of English and a grapholect (written version.)
For example, it is much more common to end a sentences with a preposition in spoken English than in written English. To demonstrate, it is very common to hear "that is one of the things I have a problem with" in spoken English and its numerous dialects. However, in written English, the proper phrase would generally be "that is one of the things with which I have a problem."
Now, all of that said, even the standard "rule" of not ending a sentence with a preposition is hotly debated among English teachers, professional editors, and lovers of language. That would get us in to a descriptivist vs prescriptivist debate, though, and I don't want to travel too far down that road.
In brief, the answer to your question is "No." The "rules" for English are a bit more lax than they are in other languages, which means that there really aren't any grammatical structures or verb tenses which are reserved solely for the written form of the language.