What is the current draw of an Arduino Due board when sleeping?
While the processor itself supports very low-power modes the LM2734 Datasheet shows a typical quiescent current of 1.5mA. It is also only a step-down regulator so the Arduino Due shows a minimum operating voltage of 6V, but you'd really want to aim higher say by using 6 cells for 9V so it continues operating once the batteries are getting low.

If you used 6 x 1.5 AAA after a quick search the highest capacity I could spot was 1200mA/hr so that would give around 1200 / 1.5 = 800 hours or about a month of operation.

But that's only taking the main regulator into account, there's also op-amps and a secondary linear regulator along with a few resistor dividers. I do not know the exact answer but as a guestimate would say it's likely to be more in the order of 5mA (10 days of operation) and not really suitable for your intended purpose.