Do the colors of the Istaris' clothing have any effects on their powers and capabilities?

No. It is the reverse. The colors of the Istaris' clothes are indicative of the powers with which they have been vested. (The same idea is that the "warning sign" does not make a band-saw dangerous, but is an indication of the band-saw's inherent danger.)

White is indicative of purity and power, which is why the White Wizard (originally Saruman and later Gandalf) is the leader of the order and has access to the far more directed and offensive (military) powers. Grey is indicative of the wayfarer, which is why the Grey Wizard (originally Gandalf) is a person who moves from place to place without having a home or distinct purpose. Brown is indicative of the ground, which is why the Brown Wizard (Radagast) is connected to earth and nature as opposed to man. Blue represents the ocean and the sky which is why the two Blue Wizards (unnamed) are distant from society and have disappeared from society.

Saruman eventually calls himself the Wizard of Many Colours which is not actually a proper wizard color. However, it is indicative of his changed state. When pure white light is broken by a prism, it becomes many colors. Saruman's purity was similarly broken by his quest for power resulting in the numerous colors.