It can be difficult to keep a dress shirt looking neat, regardless of how well it was ironed or starched before you put it on. These shirts tend to get wrinkled easily during the course of the day, especially if traveling is involved. Non-iron dress shirts were invented to solve this problem, but they have their disadvantages as well as advantages.
Non-iron dress shirts are typically made of either a synthetic material or a natural material like cotton that has been coated with chemicals to make it less likely to wrinkle or is mixed with a synthetic material to give it a little stretch. Well-made non-iron dress shirts tend to be made at least partly of cotton and use the structure of the shirt to help minimize the appearance of wrinkles, either by adding darts and making the shirt fitted or by using material with a pattern or texture that helps hide wrinkles.
The best non-iron dress shirts can be washed and worn numerous times without needing to be ironed. They can even stay wrinkle-free after being worn while traveling, a situation that leaves regular dress shirts covered in unsightly wrinkles. This means you can save the space a travel iron would take up in your suitcase and have more room for other essentials.
Non-iron dress shirts tend to be a lot more expensive than regular dress shirts, and some of the less expensive options can be very stiff and have heavy chemical coatings or be made of synthetic fabrics that don't breathe well. If shirts are not tailored well, the seams might not lay flat and the collars might be flimsy and not stand up properly.
Not all non-iron dress shirts are created equally. Some still wrinkle even on the first wear or after the first wash, while others can last as many as 30 washes before you need to fully iron them. Even the better non-iron dress shirts might need light ironing from time to time. These shirts are wrinkle-resistant, but not totally wrinkle-free.