Tires make such a huge difference to your ride. It is best to go with a pneumatic tire or one that requires air because this will provide the most cushion and reduce the side effects of sidewalk cracks, curbs and other bumps that are a part of life. Most people today are choosing the Schwalbe tire upgrade for their everyday chairs because they are a pneumatic tire but have a protective insert to prevent flat tires. Flat free tires or inserts are good but the downside is that they are heavy and pretty solid so bumps are tough but on the upside, they never need air.
If you intend to use your chair on a lot of unpaved terrain, then choosing a tire with some tread is a good idea. Just think about where you will use your chair the most because the more the tread, the more dirt that collects and guess where it ends up? Yes, you guessed it...inside.