It's a bit unclear what you're asking:
- Is this about a laser thermal printer, using paper made to work that way, or
- is it about using 'plain' paper to be burned?
- Is it about the use of a laser, or
- what other kind of device?
To start with, there are only two ways to transfer heat in a focused way and that's direct contact of some heating/heat transfer element and a beam focusing, which in turn makes most sense by using a laser. I would assume the question tries to avoid contact based printing (which would as well be possible), so laser it is.
Even for thermal paper, the necessary power to 'burn' it in a time short enough to work as a printer, is way above what lasers could do until very recently. Or more exact, could do within an acceptable budget and size. Before high power laser diodes, it would have taken a quite large setup as well.
But even with diode lasers a setup would be far larger than what existing thermo printer need.
To burn coated paper (aka thermo-paper) with a speed equal to a cheap thermal printer, a power close above .1 Watt is needed. After all, it does not make any sense to spend more money or a slower printer. According laser diodes have still today a price tag of 10 Euro or more.
Doing the same with 'plain' paper requires either a laser many times the power or to accept slow operation. For example a .5 W laser may 'engrave' under optimal setup up to 100 mm/s on conventional (laser printer) paper. Not really anywhere near acceptable speed. To even come close to a thermal printer it would need a power past 5 Watt.
Power over time is also a big hurdle to overcome is the paper catching fire vs print speed. This goes hand in hand with focusing. To make it viable the paper would need to be transported in a well defined way - maybe exactly as with a thermal printer.
In addition, there is no 'plain' paper. Even what we would consider similar may be different and designed to purpose. Some may remember that in the early days of laser and inkjet printers one had to take care what paper to use, as sheets made for typewriters did not work well wit non impact technologies. So, even if someone would build a laser based printer, 'burning' results may greatly vary over paper type used. It would only work good with paper made to certain specs.
Bottom Line: It did never make any sense, neither technological nor financial, to build such a printer.