Zinc-air cells have an awesome energy density (better than lithium-ion), on account of the oxygen being effectively 'free'. They're also cheap, and don't contain exciting chemicals. But they self-discharge at a very high rate, and a typical hearing aid battery will be dead within a couple of weeks even if you don't draw current from it, hence the stickers.
What counts as dead will depend on the hearing aid, both in terms of how much current it will draw, and what the lowest voltage it will operate at. I'd have thought most modern aids would produce annoying beeps as a low-battery never-quite-enough-warning, before conking out. Older analogue aids would just sound progressively worse as the voltage drops. It's entirely possible that the user's deafness precludes noticing any of these effects, hence instructions to change them preemptively.