Iago is constantly talking about women in derogatory ways, from his joking assessment of women for the amusement of Desdemona in the second act, to his put-downs of Emilia, to his assumption that Desdemona will probably cheat on Othello eventually. Does the play support Iago's opinion of women or refute it?

Shakespeare allows Iago, as the villain, considerable license to say things about respectable women that Elizabethan audiences would have understood as indicative of his disturbed personality.
This is most evident in his condemnation of Desdemona. The relevant information to convey that he is framing her is put right in front of the audience, both in his words and in his talking with Emilia about the handkerchief. Desdemona herself is almost unbelievably perfect, an icon of virtue. There is no ambiguity--to the audience--about her fidelity. That is part of what makes it a tragedy. We know Othello is wrong but we cannot open his eyes.
The question of whether Othello's reaction would be wrong if she had cheated on him is another matter. Iago's fantasy that Emilia was unfaithful is likewise presented as one reason he has become unhinged.
Whether or not these two women were faithful, they still end up dead from domestic violence. It may be that Shakespeare's public would have condoned husbands' violent retribution for adultery, but that is not the situation he presents here.

The play strongly refutes Iago's negative assessment of women as untrustworthy. First, Desdemona is always loyal to Othello and never cheats on him. It's tragic that Othello is manipulated into believing such a good woman has betrayed him.
Second, Bianca shows her feelings for Cassio when she cries at seeing him wounded. Bianca feels compassion for Cassio, while Iago kills him cold-bloodedly.
Finally, Emilia only turns on Iago when she realizes that he has caused Desdemona's death, at which point Iago literally stabs her in the back. However, Emilia has shown her mettle in speaking the truth. She has shown herself loyal to truth and to Desdemona. As she lays dying, she tells Othello that that Desdemona was always faithful to him. All of these are the acts of a courageous woman of character.
Looking at the women in the play, it is clear that Iago projects his own twisted desire to betray and hurt others onto them: it is not women who have a problem but Iago. Shakespeare, who so often sees through misogyny, clearly shows in this play that the women are not at fault, no matter how Iago tries to slander them.