The specific amount of time is not specified, but we do know that the story begins in the summer, when the girls are dancing freely in the forest, and ends in the fall, with the hanging of John Proctor. Therefore, it is the passage of one season, or approximately three months that pass.
The main events happen quickly, however. It is within 4 days that 39 women get accused of witchcraft, and within 1 day that the number jumps from 14 to 39. In the heat of the summer, the passions are high, and accusations fly around.
With the cool weather of fall, there are cooler heads that prevail. Hale and Danforth realize how wrong they have been, and Parris understands his own complicity - and his danger - for having contributed to the tragedy. However, fall is also the season where things begin to die, and so it is symbolic that the only scenes in which a hanging is imminent happen in this season. Proctor figuratively signs his own death order by refusing to literally sign his name.
Fall is also when the days grow shorter. The dimming of the light is shown in Miller's stage direction in Act 3, scene i, as he talks of the light coming dimly through the jail window. As hope for the moral conscience of the town has been quashed, so has the "light" of the day.
I think that three months is a little too short of a time for all the events in the play to occur. The real Salem Witchcraft Trials took two years and Arthur Miller, the playwright, certainly had to condense this timeline for dramatic purposes. The girls were dancing in the forest, but this could have been spring time and may have been celebrating some rite of spring. Eilizabeth Proctor discovers she is pregnant about one quarter of the way through the story. Since there were no early pregnancy tests, it probably took at least two months for her to be sure of her pregnancy. At the end of the play, at least in all the versions I have seen, she is visibly pregnant. This could be a directorial choice, but we also know that it is fall. This is an important symbol in the play because fall is traditionally a time of harvest, and also a time of the decay and death of leaves. However, if the play begins in the spring, and the trials occur during the hot summer, then the deaths of John Proctor and Rebecca Nurse could have been put off until autumn, which would fit symbolically with the rest of the action. So, my guess is that the play takes place over a six month period. Whatever the case, the point is that innocent lives were lost because of the false accusation of very young girls.