3*0.130 = 0.39 Wh Over 5000 years it would amount to about 8.9 nW (nanowatt), of course assuming the chemistry to be able to sustain such time span (which is not).

If drained over a year (which is as long as the battery from the post you're responding to is supposed to last), it works out to:

390 mWh / (24 h * 365 d/Y) ≈ 0.061 mW = 61 μW

In that case, one of these diamond batteries would be able to substitute for the CR1632 and give a sealed tracker an effectively infinite lifetime (at least compared to, well, human civilization). I imagine it would be pricey, though.

It also wouldn't be able to do drive a speaker, another key purpose of a Tile. Its actual tracking is primarily passive, so it's hard to locate with just a phone. (You might be able to triangulate from multiple phones... I recall an HN article with somebody doing that a few weeks ago). The main use case is to get close enough to trigger it, then follow your ears. That's thousands of milliwatts.

I suppose it might be possible to add a capacitor that's being continually charged.

Piezo buzzers can be more efficient than that. This 95 cent buzzer sold by Adafruit is rated to 9V at 3mA: https://www.adafruit.com/product/1740

Is it loud enough? Tiles are supposed to be detectable at up to 300 feet, but that's less useful if you can't actually hear it.

I don't actually know the specs they use for that. I just figured that it would take a few watts to be heard at distances over tens of feet.