I haven't hiked any of the trail north of NY, but I'm fairly certain that if utilizing mail drops, one could hike the trail for less than $1500 if they could hike it in less than 4 months. (total doesn't include cost of gear, travel, or injury) This doesn't require dumpster diving or being a hobo either. If you get in and get out of towns there just isn't anything to spend money on once you are back on trail.
Avoid zeroes. In towns, eat, resupply, recharge, shower and wash and get walking on out of there. This can be tough when others are trying to convince you to stay.
Avoid excessive alcohol. For me hiking causes me to lose the taste for it. I know this isn't the case for most anyone else.
Prepackage dehydrated meals made at home for mail drops at post offices that are close to the AT. This takes planning, a loved one to send you boxes, and the willingness to hike bigger miles when necessary to made a deadline. It also saves a ton of time shopping and repackaging. Muesli/Oats (not single serving packs) tortilla PB/squeeze jelly, TONS of cheese, quinoa, couscous, salmon packets, beans and rice (I'll never get sick of it), #10 cans of mountain house can be divided up into 4-5 servings, these occasionally drop to $12-13 a can, some like beef stew can be used as a good base to add other ingredients to. Bulk 48 boxes of snickers only cost $28, ship yourself 24 each resupply, way cheaper than Cliff bars or buying Snickers individually. So many ways to save money on food.
For on trail resupplies, when possible avoid gas stations and hitch to a town that has a walmart or grocery store for higher quality, less expensive food.
Plan for longer stretches between resupplies. When convenient, instead of resupplying every 3-4 days, plan to do so every 5-6 days. This helps cut out unnecessary town stops, reducing temptation to spend money.
Hike more hours each day. The less time the hike takes, the less it'll cost.
These are some of the biggest money savers that I've noticed in the time I've spent on trail. I'm convinced that it can be done for $1500 or less without being a mooch but it would take a lot of self control and discipline. For others that like to sample each town along the way I think it would cost quite a bit more and as others have stated a $1000 a month would be a good baseline.
Keep in mind you do occasionally have to Treat yo self.