p16-INK4a is a part of a very important checkpoint mechanism. It's the "bad guy" in the context of aging because it induces senescence, and too much senescence leads to aging-related tissue degradation.
But senescence is important. It's one of the responses cells take when something goes wrong-- DNA damage, viral infection, telomere depletion, that sort of thing. Senescent cells have stopped proliferating. We have a word for cells that don't stop proliferating, and that word is "cancer". So, p16-INK4a is a major tumor suppressor. A universal p16-INK4a knockout would have a much harder time shutting down the proliferation of cells that had undergone DNA damage, and would therefore be much more prone to cancer. You'd have very young-looking tissues filled with tumors.
So, the headline question is why INK4a is not expressed in the heart or liver if it's so important. This is speculation on my part, but I think it's because those tissues are especially prone to being damaged by fibrosis, and a build-up of senescent cells would lead to increased fibrosis. Senescence is just one possible response to DNA damage, though. Another is apoptosis. If the senescence-induction pathways aren't active in heart and lung tissue, I'd expect the apoptosis-induction pathways to be pretty active.