What would the relationships between the princesses be: Pocahontas and Tiana?

I think they'd be very close friends.

A lot of people find Pocahontas very immature, but she carries herself maturely and handles situations like an adult, just the way Tiana does. Pocahontas is just playful, and can be childish sometimes but she knows when to become serious. I think Tiana would appreciate and admire that in Pocahontas.
Now I know Lottie and Tiana are best friends, but I don't think they would have been if they hadn't grown up together, mainly because Lottie is spoiled, a little immature and somewhat foolish. This is why I think Tiana would be closer to somebody more mature and calm.
I also think that Tiana would keep Pocahontas from doing foolish, reckless things and Poca would push Tiana to have fun with her. Pocahontas wouldn't be doing stupid things like jumping off a cliff, but she'd still be having fun and Tia would totally join her, just the way she joined Naveen in messing with the hunters.
Pocahontas is very comfortable doing her own thing, so if Tia is too busy to hang out, Poca wouldn't be mad at her or feel rejected.
They'd fight little and be very good for each other. They have enough differences and similarities, and the right ones too, to fit together as close or even best friends.

I don't think they would become best friends. Because they are very different. For instance, they have absolutely different dreams and goals in their lives.