In certain fanfiction scenarios, Kaname might use his status and power to isolate Zero from others. This would make Zero feel lonely and unwanted. And also, if Kaname were to flaunt his relationship with Yuki in front of Zero, it would be a huge blow to Zero's heart as he has deep feelings for Yuki too.
Well, in some fanfictions, Kaname might hurt Zero's feelings by being overly possessive of Yuki. Since both Zero and Kaname have feelings for Yuki, Kaname's actions to keep Yuki close to him could make Zero feel left out and unloved, which would surely hurt his feelings.