No, asteroids do not have tails. Asteroids are rocky or metallic objects that orbit the Sun, but they do not have tails. Only comets have tails.

Comets are icy objects that orbit the Sun. When a comet approaches the Sun, the heat from the Sun causes the ice on the comet to vaporize, creating a tail. The tail of a comet always points away from the Sun, because the solar wind is blowing the tail away from the comet.