I have a feeling that only purpose heat stumps serve is to artificially prevent player from doing too fast what they need to do.
I might have everything I need - heat, prefabs, people to do something, but for some reason some paper stubs somehow do not allow that.

I guess they represent hours of heating citizens cay buy and if they have extra they use them as a currency for goods and services and they are getting them for doing government-requested work. Having currency kind of makes sense, considering how bigger the city have become. However, what does not make sense is how they work:
Right now it looks like citizens are printing the heat stamps at their leisure time and them gift them to steward, so they can get them back when steward orders something done. If they have more or better chairs and maybe some toys for the children they would print more of stubs even when dying of cold. I can't figure out how that makes sense and why it's here.

If heat stamp value is fixed, for example it is a "hour of heating", it would have made sense that economy size and Steward's ability to request things would directly depend on the current heat output.
If heat stamps are a more free-flowing currency, that means that Steward's office should be able to just print them with a risk of creating inflation if spending over current heat output/needs of the city and not balanced by creation of goods.

Similar issue of artificial limiting actually affects frost-breaking. Why I can't order clearing only 3 tiles at lower price, when I don't need all 8?