No, red dwarfs do not become white dwarfs.
White dwarfs are the remaining core of helium-fusing red giant stars after they have shed their outer layers due to fluctuations in the unstable helium-burning core and hydrogen-burning shell.
Red dwarfs are not massive enough for helium fusion to ever begin, so they never get hot enough to go beyond "red." When their supply of hydrogen runs out, they simply cool and condense, eventually becoming helium-rich brown dwarfs that do not orbit another star.
However, red dwarfs consume their hydrogen fuel VERY slowly, and because of this have EXTREMELY long lifespans, theoretically approaching a trillion (that's 1,000 billion!) years. Indeed, any red dwarfs that formed near the initial condensation of matter would still exist today. Astrophysicists have placed a pretty high priority on searching for such ancient red dwarfs, because their composition should conform to the early composition of the universe and could thus confirm or falsify important predictions of big bang theory. However, their very low luminosity makes such stars very difficult to detect beyond our immediate stellar neighborhood.
EDIT: Mine is correct. A red dwarf never generates enough heat to become white. This requires the higher-core-temperature stage that main sequence go into when they become red giants. (Sorry, ronwiz, no offense.)