Verbal irony is when a speaker intentionally says one thing but means something else. One example of verbal irony, somewhat of a stretch, is at the end when Jim says their presents are "too nice to use just at present." What he actually means is that they simply can't use them at present.
Dramatic irony is when the audience/reader knows more about the outcome than the characters. This is applicable in this story only if the reader actually guesses that Jim has sold his watch to buy the combs. However, if the reader doesn't guess that Jim has sold his watch before he reveals this twist, a typical O. Henry surprise ending, dramatic irony is not really applicable.
Situational irony is the best description of irony in "The Gift of the Magi" because the audience and the characters discover the actual ending together. Situational irony describes a difference between the expected result and the actual result. Della thought she made a loving, wise choice by selling her hair to buy the chain. Ironically, Jim sold his watch to buy the combs, which Della no longer had. For both Jim and Della, the expected result (a wonderful and "useful" gift) was wonderful but not useful at all.
The further irony is that these well-intended gifts were born out of love and sacrifice. On one hand, it is ironic, considering that the narrator calls Jim and Della foolish and then wise (more characteristic of verbal irony). On the other hand, it is fitting since they are so devoted to each other. But as far as the characters and readers are concerned, it is the situation that determines the irony; not the words nor the question of who has the information first.
Della momentarily forgets that she's cut her hair when she opens the combs. Since the reader doesn't find out until Della realizes her mistake, it still should be classified as situational, rather than dramatic, irony:
And then an ecstatic scream of joy; and then, alas! a quick feminine change to hysterical tears and wails, necessitating the immediate employment of all the comforting powers of the lord of the flat.
This situational irony could also be described as irony of fate or cosmic irony. This kind of irony indicates situations in which characters are at the mercy of God (or gods) and/or Fate. Considering the references to the Bible, cosmic irony is as valid as situational irony.