Do snakes inject poison using their forked tongues?

No, the poison is injected with the fangs of the snake. No. They use their fangs. What else would the fangs be for?

The forked tongue is used to smell. When they stick their tongues out, they are collecting scent molecules on the ends of the fork. Then, they draw their tongues back in and put them in special pits in the roof of their mouth. These pits act like our noses do, and detect what kinds of molecules and how many of them there are on the tongue.

The fangs are hollow and connected to venom sacks near the eyes of the snake. When a snake bites something, it uses the muscles around the venom sacks to pump the venom through its fangs and into whatever it bites. This is if the snake is poisonous. If the snake is a constrictor, then it just squeezes tighter and tighter until whatever it is squeezing cannot breathe any more and dies.