What should I do with my blessed palms from last year?

According to the Code of Canon Law, blessed items are not to be discarded in a trash can, but treated with respect (cf. #1171).

At Mass palm branches are set apart by a blessing from the priest and made into a “sacramental,” an object that is meant to draw us closer to the celebration of the seven sacraments. Throwing them in the trash ignores their sacred purpose and treats them like any other object we no longer need.

Palm branches can be burned or buried in order to properly dispose of them. This type of disposal honors their sacred purpose and returns them to the earth in a dignified way.

Anyone can do this.

The Church will not be accepting old palm due to the abundance already collected.