No, Vega, the brightest star in Lyra the Harp (visible almost directly overhead when darkness falls tonight), will not be our next North Star. It will, however, eventually become our North Star in about 13,000 years.
Presently, Polaris, the brightest star in Ursa Minor, appears close to the North Celestial Pole and therefore serves as our North Star. The North Celestial Pole (NCP) is restless and over the course of 26,000 years will describe a 47 degree arc through the sky. Consequently, the NCP will align itself toward various stars over time.
Precessional wobbling causes this NCP displacement. The influence of the Sun, Moon and, to a lesser extent, planets, cause Earth's pole to precess over a long time period. One can observe a similar behavior in spinning gyroscopes. As the gyroscope rotates, the top repeatedly describes a circle due to precessional wobbling. Our planet's wobble is much longer, of course, so that even over a human lifetime, its effects are subtle. Because of precession, different stars will serve as north stars and the constellations arrayed along the ecliptic (zodiac) will gradually change positions. Their move about one degree every 73 years.
Polaris will remain the North Star throughout the rest of our lives and for a few centuries later. Throughout the past few centuries, Polaris has served as a North Star marker for navigators, escaping slaves and other explorers. During this time, the NCP has appeared to be drawing closer to Polaris and continues to do so, today. By 2102, the NCP and Polaris will attain their minimum separation distance of 27'. After 2102, the NCP will slowly move away from Polaris and within 2,000 years will pass close to Errai, a star within Cepheus the King. Although Errai is about three times dimmer than Polaris and the NCP won't ever be as close to it as it was to Polaris, observers in 2,000 years will likely use Errai as the North Star.
Eventually, the NCP will move beyond Errai and will point to other stars.
In about 13,000 years, the NCP will be close to Vega, the brightest star in Lyra the Harp. So, in the remote future, Vega will be the North Star. As is true with Errai, Vega will never be as close to the NCP as Polaris, it is about six times brighter. Those observers in remote posterity will certainly behold a brilliant North Star in their skies.