Well I have a background in Textile Science. I iron very very little.
Knitted garments, that is t shirts, polo shirts, jumpers etc all shred creases very easily especially when worn, therefore don't need ironing.
It is possible to get away without ironing woven cotton if hung very carefully, especially trousers. I have a couple of things that I iron evey 3 rd or 4th wash, and they look fine, but I never iron trousers or jeans.
Woven viscose and linen usually need an iron, but I have a very heavy linen top I don't iron.
When these garments are washed, the molecular chains break temporarily, and the need heat and moisture to restore them. However, all natural fibres retain moisture, and when you wear them you make them worn.
My advice is to iron woven tops sometimes, especially linen. The rest should survive fine 