Not necessarily. Depending on the manufacturer and style of shoes you pick, some shoes still offer a lot of padding and are still very lightweight.
At the same time, a lot of cushioning isn't always ideal for all flat-footed runners. It all depends on your personal preference. Some of the products here have more minimal cushioning systems, while others have a lot more. If you aren’t sure what will be most comfortable for you, try some of the shoes on at the store.