Dramatic irony occurs when the audience is aware of something that is happening unlike the main characters/actors who are oblivious to such information. In Act 3, Scene 1, Banquo suspects that Macbeth is responsible for the murder of Duncan given the fact that the first part of the witches' prediction came true. Banquo believes that Macbeth took part in making the witches' prophecy come true:
Thou hast it now: king, Cawdor, Glamis, all,
As the weird women promised, and, I fear,
Thou play'dst most foully for't...
However, instead of realizing that his life could be in danger because of the second part of the witches' prediction (that Banquo's descendants would inherit the throne which could imperil Macbeth's position), Banquo secretly hopes that the witches will be right and does not entertain the possibility of his assassination by Macbeth. The audience knows that Macbeth plans to have Banquo and his son, Fleance, killed unlike Banquo who is completely unaware of this.
As far as foreshadowing is concerned, Banquo's soliloquy serves to remind us that Macbeth is not done murdering people. By reminding us of the witches' prophecy and implying that Macbeth has killed Duncan, Banquo unconsciously foreshadows his own death because his thoughts make us realize that Macbeth will not not stop until he gets rid of anyone who stands in his way. We know that Banquo is a threat to Macbeth's position on the basis of the witches' prediction, so Macbeth will want him murdered.
Dramatic irony is when the reader/audience knows something that the actor/character does not. At this point in time, we know that Macbeth is behind the death of King Duncan. Banquo at this point assumes the same thing.
However, it is too late since Macbeth is planning on having him killed. We know this, but Banquo doesn't realize yet how evil Macbeth truly is. Macbeth asks many questions trying to find what Banquo's route will be.
When Macbeth tells him "Fail not our feast," Banquo replies that he won't miss it. That could be considered foreshadowing since Banquo does appear, but he appears as a ghost.