A couple of thoughts off the top of my head.
1. Maybe it's like wearing clothes, after a few minutes you don't notice them anymore because your sensory system acclimates. Well, your heart is always beating.
2. Speaking of the sensory system, what's it like inside the chest anyway? Is it like the brain, which has no pain receptors to speak of? (This isn't rhetorical, I really don't know).
3. Sometimes I do notice my heart beating.
4. I'm not sure why the pressure is important. Do you mean because it's a strong contraction? Besides, what's the touchstone for a pressure we do notice vs. one we don't?