These are different questions, though. In the end, I think human language is in fact finite. For one thing, infinite is a boolean: either something is infinite or it isn't. Even if a theoretical dictionary has 3 billion words in it, it's finite. For another, if you look back on the history of language, you just don't see a great deal of "advancement" in terms of linguistic complexity. Some languages are more complicated than others, and ones which are unrelated to your native tongue are harder to learn than those that have some relation, but there's not really a sense that I see of, for instance, ancient Sumerian and Greek being any less complicated than modern Farsi or Greek. Dialects like American English complicate this because it borrows from the vocabularies of many languages at once but structurally it's not necessarily any more complex than languages that sit all by themselves like Basque.
I will say that writing has an effect, as does mass media, but I'm not convinced either has the effect of making things *more* complicated. In fact, I think mass media all but eliminated dialects, which makes the language less varied and easier for a guy in one town to understand what another guy in another town is saying. The bottom line is this: any living language more or less has to be understood by the average 5 year old and mastered by the average 20 year old. Otherwise it doesn't work and people adapt it to set it to those standards.