And in most countries, you'd just add that to the tax return information they send you. And if that furnace gets you discounts over years, you should only have to do that once - the next years, they'll have the information.

Or alternatively: They can directly subsidize high efficiency furnaces, which would actually allow more people to have them than a tax credit will since it would lower the barrier to entry (price).

Sure they do. They would send you a bill, you'd say, "hey I bought a furnace", they would deduct from the bill and you'd pay the rest.