to push the Kleenex across the desk

To pretend you care about someone’s loss by making a perfunctory or meaningless gesture. This comes from the common experience of a Human Resources manager pushing a box of tissues across the table to a employee he/she fired.

Jimmy: Damn her. She told me she doesn’t want to be her girlfriend, but we that we can still be friends.
Tyler: Wow, it was pretty fake of her to push the Kleenex across the desk.
Jimmy: Yeah, she doesn’t care. She’s blowing some other guy.