Why does the Hare Krishna mantra seem to have no affect on me at all?
Various factors determine why we may feel or not feel the effects of chanting Hare Krishna. One thing is that to chant with expectations of feeling something is not the most recommended approach. The process is direct and simple; chant the Hare Krishna mantra while trying to become free from the offenses to the holy name. Krishna is His name, and chanting is the most effective way to experience His presence in this day and age. Sooner or later, we will experience Krishna in His name, by His mercy.
Whether this happens sooner or later to some degree depends on the quality of our chanting. Chanting a fixed number of rounds each day, in an environment that's as free from distraction as possible, giving full attention to each mantra, can help us. Chanting in the company of others who are serious about their own chanting will also help us. Chanting while watching television, driving, surfing the Web, reading, or anything else simultaneously will not help us.
Many people—when beginning to chant Hare Krishna or even after years of chanting—find, as you have found, that the mantra seems to have no effect. Chaitanya Mahaprabhu Himself spoke of having this very same condition. He lamented in one prayer, "even though Krishna makes Himself personally available in each of His unlimited numbers of names, I'm so unfortunate that I have no attraction for them."
One thing to do is to chant while praying, "I am so unfortunate that I have no attraction for Your name. Please allow me to chant attentively so that I may experience Your presence."