Why is the spigot on my oil heater dripping?
Is it the spigot, or is there leakage from a corroded boiler?
If you determined it's the valve that's leaking, you'll need to:

Shut the electricity to the heating system to prevent it running while dry.
Shut the cold-water feed to the boiler.
Drain the whole heating system through that valve... you could run a garden hose to the fitting once it's cooled.
Replace the washer in the valve, just like you would a faucet valve.
Close the repaired valve.
Open the valve to the cold-water feed slowly.
Check for leaks.
Once the system is filled at bottom, purge air from baseboard convectors or radiators. This is a bit of a nuisance, since each valve at each heater must be opened until air stops hissing, and water starts to drip. Failure to do so will restrict heat to some areas of the house. Purging may need to be repeated after the system runs for a while!
Turn the electricity on again.
Turn up the temperature at the room thermostat and check that heat is coming out of all the convectors/radiators.

Yep, what a pain it is to replace one washer.