I am looking for a means, by RAW or RAI, to generate heat that can be used to provide lift for a balloon, or to power mechanisms that could generate lift, while not consuming fuel. While this most likely points to a spell or a magic item, I don't know which ones will allow me to make a hot air balloon (or something close to it).
Possibly the magic item Candle of the deep, a "common" magic item from Xanathar's Guide. This says it "gives off light and heat like a normal candle" — and it is unclear if it is consumed like a normal candle. Given that its major property (it burns underwater) is pretty boring in a world with the Light cantrip and the Continual Flame spell (the result of which is effectively a permanent torch which "cannot be smothered or quenched" so also presumably works underwater), it seems pretty reasonable that this magic item is not consumed. However, this would definitely be up to your DM — it's also reasonable to say "nah, it's just like a regular candle but works underwater".
Of course, even if they're not consumed, you'd need quite a few to provide lift for a balloon, but it is "common" ... so, getting a lot of them may be in the realm of possibility.
Create Bonfire might work
Create Bonfire creates a self-sustaining magic fire that lasts for 1 minute. However, as it is a cantrip, you can repeatedly re-cast it to keep the fire going longer. It can deal fire damage and ignite flammable materials in its area, so it definitely produces heat. However, it continues to burn even if no flammable materials are placed in its area. The spell has no material components. As long as you don't intentionally place any flammable materials (or flammable creatures) in the bonfire's space, this spell seems to satisfy all your requirements.
Create Bonfire might not work for a moving vehicle
Since you have added your reason for needing this spell, Create Bonfire may not be what you want. The bonfire created by the spell is stationary, but "stationary" is not clearly defined, leading to a number of questions about how certain spells work on moving vehicles, such as this one and this one. You should ask your DM how they will apply the rules in this case (or if you are the DM, you'll need to make a ruling).
Conjure Elemental might work. Given a 10-foot bonfire as raw material, it is able to summon a CR5 Fire Elemental for an hour, during which time it might be able to heat the air in your hot air balloon.
Additionally, the Eberron campaign setting uses bound elementals as the sources of motive power for their airships, so this sort of thing has precedent in official DND settings.