Other answers have well covered many aspects of irony in this scene. The most important point to keep in mind is that the overarching aspect of the scene is its dramatic irony, which is when the audience knows more than a character does in a work of literature. We know, as Duncan does not, that Macbeth has received a prophecy from the three witches that he will be king of Scotland. We also know, though Duncan does not, that Macbeth is laying plans to murder him. Our insider knowledge is what makes so many of the utterances in this act ironic.
For example, Duncan states that he finds it impossible to read what is in a person's heart from their face. He then says, about the traitorous thane of Cawdor, who has just been executed: "He was a gentleman on whom I built/An absolute trust".
For such a short scene, act 1, scene 4 of Shakespeare's Macbeth abounds with irony. At the beginning of the scene, Duncan makes a remark to his sons and others about the traitorous Thane of Cawdor:
DUNCAN. There's no art
To find the mind's construction in the face:
He was a gentleman on whom I built
An absolute trust. (1.4.13-16)
Macbeth enters at that moment.
The audience knows that Duncan has given the Thane of Cawdor's title and possessions to "worthy Macbeth," and the audience also knows that the new Thane of Cawdor is no less treacherous and traitorous than his predecessor.
A little later in the scene, Duncan announces that he has selected his eldest son, Malcolm, as heir to his throne:
DUNCAN. ...Sons, kinsmen, thanes,
And you whose places are the nearest, know
We will establish our estate upon
Our eldest, Malcolm, whom we name hereafter
The Prince of Cumberland; which honor must
Not unaccompanied invest him only,
But signs of nobleness, like stars, shall shine
On all deservers. (1.4.41-48)
Shakespeare uses Duncan's passing reference to "stars" to turn Duncan's turn-of-phrase against him:
MACBETH. ...That is a step
On which I must fall down, or else o'erleap,
For in my way it lies. Stars, hide your fires;
Let not light see my black and deep desires... (1.4.55-58)
Later in the play, Shakespeare revisits one of Macbeth's words: "o'erleap."
The word appears only once more in the play, in Macbeth's "If it were done when 'tis done, then 'twere well/It were done quickly" soliloquy at the beginning of act 1, scene 7, in which Macbeth explores a number of reasons for not murdering Duncan.
At the end of the soliloquy, Macbeth indicts himself for having no other reason for murdering Duncan and taking his throne than pure ambition:
MACBETH. ...I have no spur
To prick the sides of my intent, but only
Vaulting ambition, which o'erleaps itself
And falls on the other— (1.7.25-28)
Macbeth does, in fact, "o'erleap" himself, and fails in overcoming Malcolm and his own ambitions.
Dramatic irony is when the audience knows more about events and situations than the characters or actors in a play, movie, or story. In act 1, scene 4 of Macbeth, Shakespeare utilizes dramatic irony when King Duncan affectionately addresses Macbeth as his "worthiest cousin" and praises Macbeth for his valiant efforts in battle. He is also excited to travel to Inverness and is anxious to dine with Macbeth and his wife. The audience is aware of Macbeth's ambitious nature and knows that he has serious thoughts about murdering the king. However, King Duncan feels comfortable visiting Macbeth's castle and is completely unaware that Macbeth is thinking about assassinating him.
Another example of dramatic irony concerns Macbeth's positive, humble responses to King Duncan. The audience is aware that Macbeth is blatantly lying to the king when he tells him, "The rest is labor which is not used for you" (Shakespeare, I.iv.45). In actuality, Macbeth will not be content until he has obtained the title King of Scotland.
In Act 1, Scene 4 of Macbeth, King Duncan meets Macbeth for the first time since the great battle. Duncan expresses his boundless gratitude for Macbeth's indispensable help against the enemy and concludes by saying, rather ironically:
Only I have left to say,
More is they due than more than all can pay.
This is ironic because the audience knows full well that Macbeth is thinking of taking everything away from Duncan. It is almost as if Duncan knows Macbeth's intentions and is unconsciously giving him permission to do it.
Macbeth's reply is loaded with irony. He tells the King:
The service and the loyalty I owe,
In doing it pays itself.
Macbeth is saying just the opposite of what the audience knows he is thinking.