Heres a few things that is against Zutara
1 Zuko is joining them because of Katara, he's there to help Aang
2 He has no feelings for Katara
3 He doesnt know her name
4 Katara would not join Zuko to help him find his mom Her main prioity is to help Aang she is his waterbending teacher,If she were to somehow go, the others would go too I dont see Sokka and aang letting her go by herself with Zuko
5 There isnt much time left
6 Zuko will also be looking for Iroh
7 There is also the Mai factor, no matter how you think about it, he has feelings for Mai you cant diss that
8 The trust issue, Katara and Sokka arnet just going to trust him right off the bat
9 Aang and Katara have to work out what is between them. Whether they get together or not I just dont see Katara jumping into Zuko's arms the minute he joins the group
10 Zuko is going to be teaching Aang how to firebend
I also think that Zuko uis going to help Aang with Katara. He's going to realize that there is something going on between them.