Summing together this wiki page (which I've now updated to reflect this answer), this reddit list, and @MageXY's answer (which I've verified from this video), I've found six characters that interact with Bastion: Lucio, Mei, Zarya, Zenyatta, Reaper, and Torbjorn:
Lúcio: Hey, Bastion! *imitates beeping*
Mei: Bastion? You will make the perfect research assistant.
Zarya: I have destroyed more of your kind than you want to know.
Zenyatta: Tell me your thoughts, my friend.
Reaper: (When killing enemy Bastion or Zenyatta) Tin cans: Dime a dozen.
Torbjorn: I hate working with these talking tin cans!