Protestants would consistently reject certain things connected to the doctrine of Purgatory. Regarding justification by faith alone: the fate of the dead should not be affected by their own suffering (Christ's atonement being sufficient) nor by the prayers and other works of the living. Moreover, Protestants would not believe in a role for the Church here - in granting indulgences and so forth. All of this comes from the "usual" Biblical basis for those Protestant doctrines, without specifically addressing Purgatory.
If Purgatory is seen only as a place where post-mortem sanctification happens, things are a bit murkier. That is, if we imagine it as merely a necessary cleansing step before dead people can enter heaven, then is there any Biblical argument against it? (Other than the lack of an explicit argument in favour, of course, as the Articles of Religion say: "a fond thing vainly invented, and grounded upon no warranty of Scripture.")
Protestant thinkers have tended to argue that sanctification is completed or consummated by the moment of death. John Wesley, for example, taught that the perfection we seek in life is completed at "the instant of death, the moment before the soul leaves the body" (Brief thoughts on Christian perfection, 1767). If this is so, then there may be a "very brief transition from death to paradise" (The Last Things, Donald Bloesch, 2004) but not a prolonged period of transformation.
This is backed up by such Biblical references as 1 Corinthians 15, where the raising of the dead in their already-perfected bodies is essentially instantaneous, and Jesus' words to the thief on the cross, "Today you will be with me in paradise" (Luke 23:43). N T Wright points to Romans 6:6 as indicating that the death of the body entails the death of sin (For all the saints? Remembering the Christian departed, 2004), and, more importantly, to Romans 8:38-39 for the vivid statement that nothing, even death, "can separate us from the love of God in Christ Jesus our Lord". Surely, he says, there is no separation for the saved after death.
However, these Biblical texts are not absolutely decisive about timescale. Perhaps time is not even a sensible concept here, as in more recent Catholic thought where specific periods of time are not mentioned (Indulgentiarum Doctrina, 1967). But the objections to Purgatory as a place of punishment, and to indulgences remitting punishment, remain even if some kind of post-mortem transformative sanctification process is accepted.