I think it is a very individual thing? No way of knowing if your child will develop health issues later on, but statistically I believe it is more likely for prem babies.
My neice and nephew (twins) were born at 30 weeks. They have asthma, but so do all their siblings (including their sisters who are another set of twins born at 36 weeks and one other sister) and more importantly, so do their parents. I would think family history is a big factor here too.
My neice (the 30-weeker) has severe immune deficiency, but as far as I'm aware this isn't attributed to her early birth.