The water side of the heat exchanger can become clogged by debris in the water, corrosion of the heat exchanger walls or both. You won't get to see this kind of clogging, so we won't talk about it much.
Soot may also clog the fire side of the heat exchanger in the exhaust flue passages. This is usually because of poor burner adjustment, a defective burner or a lack of maintenance.
Rust can clog the fire side of the heat exchanger, too, usually from condensation.
This can result in reduced efficiency of the heating system. A soot buildup on the heat exchanger, for example, restricts the heat transfer, resulting in more heat going straight up the chimney. It can overheat the heat exchanger if the exhaust flow across the heat exchanger is restricted. In severe cases, it can lead to spillage of exhaust products back into the house through the burner.
With a mirror and flashlight, look for black, sooty deposits on the heat exchanger. These should not be seen at all on gas burners and, although some soot can be expected on an oil burner, watch for measurably thick buildups. Look for rust on the heat exchanger that can completely obstruct the fins on a copper tube boiler.
Check for spillage of combustion gases as you would on any burner. One cause may be a restricted heat exchanger passage.
Where you have identified a partially clogged or heavily sooted heat exchanger, you're probably looking at a maintenance item, rather than a replacement item. This is a far less serious condition in most cases than a leak or severe rusting of a heat exchanger.