Heating
The traditional way to heat a cave is with an open wood burning fire. This can be a novelty for a while but they are grossly inneficient with most of the heat going up the chimney. They can also be smokey and dusty. A far better option is an enclosed log burner or estufa and if you can afford a little more a multifuel Dovre fire is the ulimate in fires. They are very efficient, throw out up to 12kw of heat, clean and ecological. Additionally their design allows you to carry heat, via ducting, to other rooms.
Cave rooms do not require much heating but they do need some heat in winter for you to feel comfortable.
Central heating is rarely found in caves but you should consider this if you are restoring a cave home and have the opportunity and the room to install the pipework and boiler systems. Both heating oil and LPG gas are available as fuel.
And excellent and very simple alternative is to install high efficiency electrical radiators. These are reputed to be as cheap to run as oil and have the added advantages in that they are easy to install, inexpensive to buy and can be installed only in the rooms that you want to install them in.
Ventilation
In the old days cave homes only had one door and that was the one leading outside. Internally for privacy cave rooms were separated by heavy curtains and doors were not used. This meant that internal rooms (without windows) were ventilated because the curtains allowed plenty of air past.
Today, we like to see doors installed for privacy, warmth and quiet. This can cause some problems with ventilation, condensation and dampness in internal rooms. The intensity of the problem varies from cave to cave, some caves are naturally drier than other. If you are restoring a cave you should aim to install some forced ventilation in internal cave rooms with a fitted door but no window.