The entire storyline of Othello is precipitated by Iago's resentment over being passed over for a promotion in favor of Cassio. That being said, why doesn't it seem to occur to Othello that Iago might have reacted this way?
Something about the way the question is phrased bothers me, so I’m going to start this answer with a discussion of ‘why’ questions (of which there are many here at Literature Stack Exchange).
This kind of question asks for an explanation of the actions of a character. The implication is that the actions are sufficiently implausible or ‘out of character’ that they require explanation. This can be reasonable in cases where the character is long-established so that there is an independent body of knowledge about how they behave, against which to evaluate their actions and determine whether they are ‘in character’ or not.
But this is nothing like the situation in Othello. His promotion of Cassio over Iago is more or less the first thing we learn about Othello. We’re not in a position to hold Othello’s character constant and consider his actions in that light; instead we have to observe his actions and consider his character in their light. Our question ought not to be, ‘why did Othello do this thing?’, but rather ‘what does this thing tell us about Othello?’
Considered this way round, the evidence of the play is that Othello is a poor judge of people, and prone to acting on impulse without considering the consequences.
He promotes Cassio over Iago without considering the effect on Iago.
He marries Desdemona in secret, even though he ought to know that this will offend her powerful father Brabantio. (In the event he is easily able to obtain Brabantio’s consent, so it would have been politic to have done this first.)
He dismisses Cassio from his service after the fight with Roderigo and Montano without hearing an explanation. (If he had gone into the business, he would have heard that Roderigo had provoked the quarrel, and might have wondered why.) This is sure to create an enmity with Cassio and his supporters.
He is easily fooled by Iago into believing in Desdemona’s infidelity, and doesn’t take independent steps to test this conclusion.
But with regard to the promotion of Cassio, it seems to me that Iago overreached himself. Iago’s strategy is to flatter Othello and paint himself as honest and loyal. He is all, “My lord, you know I love you” and Othello falls for it. But this strategy backfired because it led Othello to believe that Iago is completely dependable and subservient, and so his pride wasn’t going to be hurt by the promotion of Cassio over him.
It seems that Iago is so enamoured of his own deceit that he also fails to consider the consequences of his actions. A little forethought ought to have told him that if the business with Rodrigo, or with the handkerchief, ever came out, then he would be in a very awkward position.
Iago isn't entirely consistent in his motivations. He also blames Othello for sleeping with his wife, something he says in soliloquy, which is usually treated as truthful. Iago's primary mention of a motivation against Cassio is in a speech to Roderigo, to whom he lies on more than one occasion. Iago's motives are famously unclear.
Which is to say that Othello may simply not have noticed them. People are passed over for promotion all the time without turning to complex, Rube Goldbergian revenge plots. Iago acts as Othello's friend, and Othello may simply believe that Iago is a good soldier who accepts the chain of command. Othello may even believe that Iago is motivated to show good discipline in order to receive promotion next time.
That said, Othello is remarkably obtuse. Iago is aided by a series of coincidences. He had absolutely no reason to hope that Cassio might engage in an entire scene about his love for Bianca without once mentioning her name, which would have spoiled the whole plan.
There are many ways for an actor and director to resolve that tension. But I would say that most of them will revolve around Othello's genuine compassion for Iago, and Iago's remarkable skill in convincing Othello that the friendship is returned. That, and simple disbelief that Iago would engage in so convoluted a plot.
The question reminds me of the “why didn't character x act rationally” questions were discussed on the meta site a few years ago. It also reminds me of the type of discussion that Terry Eagleton talks about in the first chapter of How to Read Literature (Yale University Press, 2014): readers talk about characters as if they were real people and someone overhearing their discussion (Eagleton uses Catherine Earnshaw and Heathcliff as examples) wouldn't guess that they were talking about a literary work.
If we ignored the literary context of this question, we could counter it by asking, "Why would it occur to Othello that Iago, and only Iago, might be upset about being passed over for promotion of lieutenant?" After all, Othello has many officers and he only promoted Cassio. Surely you can't expect a general to worry about the hurt feelings of each of his officers? The army, a machine for killing and maiming people, does not work that way.
But these are all non-literary considerations, since we don't hear any other officers complain about Cassio's promotion. Characters have no backstory besides what is mentioned in the play. We don't know what the girlhood of Shakespeare's heroines was like or how many children Lady Macbeth had. This type of character analysis, which was popular in the 19th century, is completely outdated now. (And psychoanalytic criticism is an entirely different beast.)
Moving back to the play: it doesn't occur to anybody that Iago might feel slighted because he doesn't mention it to anybody except Roderigo (Act 1, scene 1). Iago fools all those around him about his intentions except Roderigo and his wife Emilia, both of whom need quite some time before they become suspicious enough to say something that might give the game away.
Readers who think that Othello is easily fooled should take into account that the general is not the only character who calls Iago "honest" or "good": Cassio does the same in Act 2, scene 3 ("Good night, honest Iago"; notably after Iago got him drunk in order to provoke the fight that gets Cassio demoted!) and Act 3, scene 1 ("I never knew / A Florentine more kind and honest."); Desdemona addresses him as "good Iago" in Act 4, scene 2.
Another factor is that nobody except Iago considers Cassio unfit for the position to which he gets promoted. For example, when Lodovico arrives in Cyprus with a message from the Duke of Venice in Act 4, scene 1, it appears that "they [the senate of Venice] command him [Othello] home, / Deputing Cassio in his government." In other words, the senators consider Cassio up to the task of taking charge of the military on Cyprus.
It is true, as Gareth Rees has pointed out, that Othello "does not take independent steps" to check whether Desdemona is faithful to him. Instead, he responds to the situation as a general: he quickly delegates the task to an officer, i.e. Iago. In other words, the qualities that serve him well as a military leader work against him when unthinkingly transferred to the private realm, where this delegation is inappropriate. But why should he suspect Iago at all? Iago always appears to defend those who appear to be at fault. In Act 2, scene 3, he pretends to defend Cassio after the drunken brawl: e.g. "I had rather have this tongue cut from my mouth / Than it should do offence to Michael Cassio". Similarly, in Act 3, scene 3, the central manipulation scene, he defends both Cassio ("For Michael Cassio, / I dare be sworn I think that he is honest.") and Desdemona ("She may be honest yet." even though he admits the handkerchief speaks against her).
Norman Sanders points out in his edition of Othello (New Cambridge Shakespeare, 1984) that critics of the play fall into two camps with regard to the main character: those who defend him as the "noble Moor" and those who condemn him as an easy victim of Iago's manipulations. Apparently, these views are not exclusively based on the play itself:
It is obvious that the two main views of Othello are based as much on the ideals of the critics themselves as on their reading of the play.
The view that critics have on Iago is linked with their view on Othello. Those who defend Othello see his fall as evidence of Iago's expert manipulation skills; those who condemn him regard Iago as a "miscalculating improviser" (Sanders, p. 25). Whatever view one takes, Othello would have been an entirely different play if its title character had not blindly trusted Iago, i.e. not the play that Shakespeare had in mind.