I have a hot water baseboard oil fired system. I am going away for a while and would like to shut off the cold supply to all the house in case of a water leak. If I leave the furnace on to heat the house, will shutting off the cold water supply to the furnace be a problem?
It should not be a problem. Your boiler should have a cut-off switch so if, for example, a baseboard springs a leak then the pressure drop will turn off the boiler. If your boiler is old enough to not have such a switch, then turning off the water is probably a bad idea.
I think it is somewhat risky to shut off the water supply because you have deviated from the normal. At least you might want to test your plan before you actually leave.
I was told by my oil delivery company that my baseboard system is a closed one and does not replenish itself. I have not had the time to investigate the system fully, including whether it had ever had antifreeze added, but I will get to it eventually; two years into this house, and that is still a low priority project.
That said, I did believe the oil company enough to shut the water off to the house, and to the boiler, each time I have traveled since moving into this house, including a recent two week vacation. I have seen no deleterious side effects from shutting the water off. My advice is to call whomever services your boiler and ask them; as I did.