Thanks Edwin,

If I understand this right, the thermocouple is a sensor which senses
if the pilot flame is hot enough to sustain burning the gasfuel from
the burner, right?
So if it's broken, the thermocouple keeps the main gas valve located
in the pilot assembly closed and it shuts off the gas valve to the
burners?

Hm, that makes sense and that is possibly a reason why my pilot is on,
but burner is not.

Is it hard to replace? Where would it be located on this 20 year or
gas furnace, brand: CLARE?
Any idea? Should I do it myself or is there a way to just clean the
thermocouple from dust or something like that to make it work?