Why doesn't Othello investigate Iago's actions?
Shakespeare suggests several reasons for Othello's trust in Iago. First, Iago has been trusted by many in the past. If this were not the case, then he would not have the position in the military that he has when the play opens and he would not be afforded such close relationships with influential leaders such as Othello. In fact, Othello places such complete faith in Iago that he entrusts Desdemona's safe passage to Cyprus to Iago.
Secondly, Iago is a master manipulator. For years, he has been able to display a facade that all seem to believe in. His wife, though she spars verbally with Iago frequently, still wants to please him and secures the handkerchief for him. So, the person who should know him best--Emilia--does not even discover until the play's end what her husband has been doing. Likewise, Iago has obviously schemed in detail for years. He knows which characters he needs to fulfill various roles in his plan, and he knows what their weaknesses are ahead of time so that he can ensure his own success.
Finally, Othello does originally question Iago's statements. As Iago talks to Othello in the garden and suggests an adulterous relationship between Desdemona and Cassio, Othello demonstrates disbelief and eventually asks for ocular proof (tangible evidence). Even after Iago is able to identify Cassio's possession of the handkerchief as "proof" of his relationship with Desdemona, Othello is not ready to act, but Iago's consistent "suspicion-planting" wears down the once savvy general.
By the time Othello could have investigated Iago, it is too late.