I have not been laid off and still have benefits, but since there is no work, I have to take time off without pay - indefinately. I can't apply for unemployment because I still have a job. Is this legal in Arizona? How long can they keep me officially employed without paying me (I am a salaried employee)?

No, you CAN file for unemployment and you should; you aren't working. You don't have to be officially "terminated".

If by "salaried", you mean "exempt", you do not have to be paid your salary for any full workweek in which you perform no work.

The AZ unemployment application requires a "date last worked."
If I put today, I am sure my employer will object to the claim.

Just because the employer objects does not mean that the state will deny your claim.

Put whatever your last day worked actually was.

Agree, you put the date you last actually worked & not the current date just because you haven't been officially terminated yet.