Dull mower blades can mangle the grass leaving it torn and bruised, which exposes it to pathogens. The open exposed area makes the grass unhealthy as diseases and bacteria can enter. Unhealthy and diseased grass will usually require more fertilization and pesticides to gain a healthy state again. Having a sharp mower blade is important, as it will help to reduce the chances for any lawn disease, which reduces the amount of chemicals used, and saves time and money in the long-term.
Here is a great article on how to sharpen a lawn mower blade yourself: