Let me take this bit by bit.
Is the Faith of the Seven a false faith?
If we mean are the Seven actual supernatural entities or just myths, then we still don't know. We don't even know if R'hllor exists or not. Sure his followers have exhibited some magical powers, but so have others like the Warlocks of Qarth who don't subscribe to that faith. R'hllor may be just a myth that's used as a name for an unexplained force.
Why does the Faith of the Seven have little presence in the HBO series?
Because it is ubiquitous. The vast majority of the people in Westeros follow the Seven, and only a minority worship the Old Gods. The followers of R'hllor are smaller still. It's like watching a movie set in America. Christianity, being the majority faith, barely makes a presence. But if a character is, say, a Buddhist it becomes much more apparent in contrast.
Does the Faith of the Seven have no impact on the story?
Absolutely not. While the actual Seven have yet to make an appearance, their followers wield tremendous political power. Aegon I had to convert to the Faith of the Seven to curry their favor. His two immediate successors Aenys I and Maegor I had to fight a bitter and prolonged war against the Faith after they rebelled. It wasn't until Jaeherys I named the Iron Throne as Protector of the Faith did the war end.
Spoilers for A Feast for Crows and A Dance with Dragons:
With a new, and fundamentalist, High Septon and a newly resurrected Faith Militant the Faith of the Seven is more powerful than ever. They have already taken down Queen Cersei and have put Queen Margaery on trial (after imprisoning her for quite some time). The Iron Throne and the Small Council no longer has the political clout to stop the High Septon from doing what he wants. And from the looks of things, the Faith Militant is set to clash with King Stannis whom they see as a heretic.