Transformer's oil circulation
Using a pump may cause cavitation in the oil, i.e. bubbles. Those could indeed significantly change the heat transfer properties of the oil. Contamination, metal filings, from the pump mechanism can also be problematic.

However, to say the oil does not circulate is an error. 

In an Oil-Immersed Self cooled Transformer the physical construction of the plumbing causes the oil to circulate by convection.



Because hot oil is lighter than cold oil, it tends to rise inside the core. At the same time the oil in the tubes is cooled by the surrounding air and becomes heavier and falls, being replaced with the hot oil from the core. As such, when the transformer is working the oil continuously cycles through the transformer.

However, there is a limit to the effectiveness of this method and larger transformers do use a pumped oil system with a secondary water cooled radiator. Presumably, the cavitation effects are factored into these designs and sufficient filtering is used to contain contaminants. The other nice thing about this type is the radiator can be remote from the transformer.



Sources.