Thrift Stores Are Not Free Dumpsters, And Using Them As One Takes Away From Charity
I get it, you don't want to haul that nasty mattress all the way to the dump (or other disposal station), then possibly pay a fee to get rid of it. It's a real pain to deal with. However, you really don't have another option that doesn't make you a complete a-hole.
Leaving it by your dumpster is a great way to get a code violation ticket. It's also super gross and rude to your neighbors who have to use the alley too.
Some folks "donate" disgusting mattresses, broken furniture, or otherwise obvious trash to local thrift stores, thinking that they will get rid of it for them. When you do that, you literally take away from charity, which happened to the Lubbock Habitat ReStore, which benefits the Habitat for Humanity:
PSA: Please do not drop off items when we are closed. Our landlords with Melonie Square frown on it, and it is very close to dumping, which is against city ordinance. Also, if it is not in very good shape, it takes away time and funds from Habitat for Humanity for us to haul it off. We appreciate you thinking of us for your donations, but we have to be good neighbors with our community as well. Thank you for your cooperation.
If you think your item might actually be resellable, then by all means donate it during business hours to your favorite thrift store. Otherwise, please use one of many disposal stations in accordance to the rules.