They are too busy to do research.In the context, what she meant is:
They are too busy doing research.Note that these two utterances are opposites. The first suggests that no research is done, while the second indicates that lots of research is done. What she meant was that the academics in that particular university are so busy doing research that they don't have time to prepare their teaching properly.
This contrast between 'too Adj to V' and 'too Adj V-ing' is rather tough for non-native speakers to get right. Note that there is no distinction between 'to-V' and 'V-ing' in the following:
I like to do research.
I like doing research.Given that (presumably) lots of resarch gets done in both cases, so there is apparently little or no difference between the meaning of these two utterances, it is little wonder that non-native speakers get confused.
Before you conclude that this is a problem with non-native speakers, so they'd better work harder to learn better English, note that native speakers also sometimes get confused. See, for example, Language Log (July 5), which discusses an instance where two journalists in the New York Times wrote "No one is too busy not to look at this" when they meant exactly the opposite.
This area of English really is fraught with difficulties. Maybe the pressure of simplification provided by the emergence of World Englishes will provide the stimulus for it to change.