No, it doesn't. "Pinnochio" is relatively known in Russia but it's not very popular. It's well overshadowed by an adaptation named "Buratino". However, aside from the general premise of having a wooden boy with a long nose as the hero it's a completely different story. For one, his nose didn't grow when he was lying. It was simply long from the beginning and he was fine with that. So there's no connection between long noses and lying in the culture.