Buddha spoke about cessation of karma and liberation from karma. For Arahant, or for Buddha, there's no acquiring of karma anymore. Indeed, if they still acquired karma, how could we call it "liberation"?
But how can an Arahant or a Buddha act without acquiring karma?
In my understanding, this is possible because only appropriated action, i.e. action considered to be done "out of my own intention""by me" counts as that person's own karma.
Since an enlightened being does not reify the concept of "a being" or "self", there is no-one left to acquire karma. There is no way Buddha could appropriate his actions as "I did this", because he had done away with the notion of "I", he only used it for communication with the unenlightened.
So whatever we think is done by Buddha or Arahant, from their perspective is not their (personal) intention, and is therefore not their (personal) karma.
Buddha's actions can still have results. Generally speaking, in terms of results, this is the third type of actions, one that leads to cessation. We can still observe some of the results of Buddha's actions in our own lives - us studying and practicing Dharma. In fact, the results are unfolding in front of our own eyes, in form of this question and answer site. But these results are not Buddha's personal karma, since they did not come bundled with the concept of "self".