What is in fact occurring is the combustion gases are cooling to quickly inside the flue and condensing. Oil is about 15 percent water and when burned the water in the oil turns to vapor. Normally the combustion gases carry the water vapor out the flue, but if the gases cool to quickly the water begins to condense inside the flue.
What can cause condensation to occur are any number of factors. A long chimney and or a large flue in combination with a newer efficient unit are most often the cause. A long chimney allows more time for the gases to cool. A large flue area can not be warmed enough by the gases expelled from the unit to create good draw to force the gases out the flue quickly.
Newer heating units are more efficient than their predecessors. This results in lower combustion gas temperatures expelled from the unit into the chimney flue. Older units stack temperatures (where the combustion gases leave the unit) were around 600º F, today they are about 450º F or lower.
Another reason this phenomenon can occur is an extremely dirty chimney flue. Many homeowners do not understand that their oil service technician does not clean their chimney. So the chimney is unknowingly neglected and over time these stains will often appear. Also the unit will be more difficult to tune becoming less efficient and consequently more costly to run.
The unit pictured had a somewhat different problem. This is a replacement for the original furnace installed in this Meriden, CT townhouse condo. The chimney is entirely constructed of metal with an integral cap. This type of chimney is usually "tuned" to the furnace it vents. So when the new furnace was installed the venting conditions changed subsequently causing condensation and the stains. It was also discovered during the inspection there was a large gap around the burner tube penetration into the ceramic combustion chamber. This may also be a contributing factor to this units venting problems.
Good technicians and installers today are aware of this problem and are addressing the issue in a number of ways. The most common is a stainless steel liner inserted inside a masonry chimney flue. The round insert is correctly sized to the heating unit to provide optimum venting.
Another option is direct venting. The furnace is vented into a short pipe through the wall. This requires no chimney of any kind and is becoming much more commonplace.
I would also recommend Steven Smiths blog on chimney liners for some further information on the topics touched on here.
So the next time you see water and rust stains on an oil fired heating system flue, it's not rain drops falling down your chimney. It means it's time to call in a professional to tune up your system.