Are arahants infallible? I've heard it said that Popes are infallible,(whether one believes this or not). 

Although the arahant has removed all the defilements, can they still be wrong, make mistakes, or are they all-knowing? I vaguely remember a story of an arahant playing a dirty trick on his less than wise brother?
Yes and no in the case of infallibility.

Infallible - what ever they do their actions do not created new conditioning. The actions are not based on the 3 unwholesome roots.

Fallible - though the actions are not based on unwholesome volition, they may not always be the optimal or best cause of action. Their infallibility is based on the fact they are never wrongly motivated, though the outcome of any action may not always have the desired or optimal results and also some action can be done differently to make a bigger impact.

They know the cause and effect regarding the mind and matter process hence understanding enough about the cause of misery and the elimination of misery.