Is disposing of old or unused clothes and belongings by throwing them in the trash considered haram and wasteful? And if so, how can one atone for this sin?

It is not permissible to dispose of usable clothes and household items, or to give them away as charity, by cutting, damaging, or throwing them in the trash. This is because it constitutes a waste of money and blameworthy extravagance, which is prohibited by the Almighty's saying: "And do not be extravagant. Indeed, He does not like the extravagant," and "And do not squander wastefully." And due to the Prophet's (peace be upon him) saying: "and wasting money."

The expiation for this is seeking forgiveness and repentance to Allah the Almighty. If something belonged to another person and you damaged it without their permission, you must seek their forgiveness or compensate for its value.