The wikipedia entry you reference describes two opposing forms of time dilation, one that will make you age quicker and one that will make you age slower. Both have noticeable effect only on extremely extended or close to light speed movement. The ISS astronauts, for instance, said to have aged slower, by 0.007 seconds for every 6 months on the station.

Consider, however, that The Flash mostly runs around on Earth. The earth's circumference is about 40,000 kilometers. Even if the Flash runs at only 13% of light speed, he can go around the entire earth in one second. As I understand it, he spends 99% of his time in "normal speed", only gearing up for full speed when necessary. This means that to achieve a sizeable time dilation will take decades, probably. Maybe more.

(Disclaimer: I did not really run any numbers here, neither for time dilation or for aggregated time spent in near light speed. I'm going mostly by intuition here)