The average washing machine drum size is about 70-odd litres (the largest that fits in a standard cabinet is 78 litres) or 0.07 cubic metres; that's 8750 boxes, but divide that in half because of packing hassles and because you want space in the drum, and you get around 4375 boxes - lets call it 4000 for a round number. So we need 31,250 cycles which take a conservative 30 mins each between the shortest wash cycle you have (and interrupting it if it's too long) and the compost sieve; that's 15625 hours, or 7.8 years working 8 hours a day, 5 days a week, for a total revenue of $1.25 million. Mindnumbing work but at an average wage of $160k per annum, not horribly unrewarding. Besides, after the first month or so, you could automate the whole process...
Alternatively, tip all the boxes out onto the ground, spread them out a bit so that they're in a single layer (touching is fine), then set fire to one corner of your layer and let the fire burn through the boxes in a wave. Just because cardboard burns at a higher temperature than pennies melt (and aren't US pennies an alloy with a higher melting temperature?) doesn't mean that there's enough thermal energy released in burning one small cardboard box (or transferred for that matter) to melt one penny... though I'd test this on one box first So long as all the pennies aren't immersed in a large long-burning mass, the heat dissipates upwards into the air and the damage should be only cosmetic.
Average cost of bonfire fines <<< $1.25 million
*ahem* Reposted because I missed rule 3 the first time reading them
(And because the math was bugging me so I went back to confirm it)