Jim and Della's useless gifts are examples of situational irony, because the results of their gift-giving are the opposite of what's expected. When Della opens Jim's gift, she finds the hair combs she's coveted for a long time:
Beautiful combs, pure tortoise shell, with jewelled rims—just the shade to wear in the beautiful vanished hair. They were expensive combs, she knew, and her heart had simply craved and yearned over them without the least hope of possession. And now, they were hers, but the tresses that should have adorned the coveted adornments were gone.
Though Della's desire for the combs hasn't been mentioned before this point in the story, Della's hair certainly has. The beautiful long hair was Della's only prized possession, and by all indications in the story, she was rightfully proud of it. However, hours before Jim brought her the combs, Della cut and sold her hair in order to afford Jim's gift. No matter how beautiful the combs are, they have no practical value unless and until Della's hair grows back.
Della takes comfort in the fact that she will be able to wear the combs again someday. But when she shows Jim the watch chain she got him—paid for with her shorn hair—he doesn't react the way she expected:
"Isn't it a dandy, Jim? I hunted all over town to find it. [...] Give me your watch. I want to see how it looks on it." [...]
"Dell," said he, "let's put our Christmas presents away and keep 'em a while. [...] I sold the watch to get the money to buy your combs."
Jim responds matter-of-factly that Della's gift is useless and might as well be put away for the time being, explaining that he no longer owns a watch to display; he sold it in order to afford the combs that Della can't use, either. This time, the disappointment is greater: while Della's hair will realistically grow back, allowing her to use the combs, Jim will probably never own a watch he'd value as highly as the heirloom watch—so the chain will probably stay tucked away.
The irony serves to highlight Jim and Della's love for each other. The audience would expect that after sacrificing her hair, Della would have the satisfaction of giving Jim a gift he'd cherish, and that Jim would respond with a comparable treasure. In a sense, that's what happens; but readers' expectations are undermined by the fact that the two gifts basically cancel each other out. What they've both bought at great cost ends up being useless.
There's an additional irony in the fact that although Jim and Della both purchased gifts that would serve as eye-catching accessories, their willingness to sacrifice for one another—reflecting their internal characters—ends up being the real gift. That mutual sacrifice means that the couple's few material riches—Jim's prized heirloom watch and Della's hair—are gone; they've even gone to waste. All that's left, the story suggests, is the couple's enduring love for one another, which can't be measured in money or possessions. The ironic overturning of expectations highlights the true strength of the couple's marriage.