The design of the HS4 carb means the oil will end up mixed with the petrol in the engine. The piston in the dash pot moves up and down. It moves down quicker than up as it's spring and gravity assisted. So when the piston goes down the oil either goes around the damper/restrictor or goes round the edges of the piston, which also helps with movement.

As you may well know when you put your foot down the cable turns the throttle disc in the carb which, much like turning on a tap, allows more air to be drawn in to the engine. As the air rushes between the bridge and piston the air pressure on the downstream (engine) side drops and this is used to suck the piston up the dashpot so drawing the needle further out of the jet and increasing the fuel flow in proportion to the air consumed by the engine. Because air is compressible and the piston has weight and therefore inertia the piston would constantly flutter up and down as it continually over compensates for itself: this vibration would actually restrict the fuel flow to the engine as well as make the car drive like a jittering jittery thing. This is where the damper comes in: a very simple open topped tube of oil with a loose fitting piston submerged in it. The resistance of pushing the oil round the piston smooths out the vibrations in much the same way as a shock absorber does for your suspension.

If you take care to keep the piston upright, the oil should last almost as long as the car without any need for topping up - as long as the oil reaches the plunger, you'll be able to feel the resistance and hear the squelching noises, there's another in there. It only takes about a tablespoon's worth to fill the whole damper and if you overfill it it will just drain down into the throat of the carburettor and make the engine smoke for a few minutes until it's all burnt away.

Everyone seems to have a different idea of what's the best to fill the damper with; regular grade engine oil is the stock answer - just like stiffer shockers don't alter a car's ride height the choice of oil won't effect the mixture under steady load but a lighter oil might slightly increase the engine response by allowing the piston to rise faster drawing more fuel and a bit more air under acceleration. Just avoid oil with a viscosity that varies a lot with temperature.

Source: The Mini Forum