The ironic gift
A living animal is a difficult gift to receive, because in Mr. Popper's case, the gift obligates him tremendously. He has to work very hard, instantly, and for a long time afterward with no end in sight, all of a sudden. The bird isn't well when he receives it, so he has to work to help the penguin heal from its bad care. He has to sacrifice for the gift, making the gift ironic. The gift is not to give Mr. Popper something, but to take things away from him by making him sacrifice so much.
The irony of the ending
The delightful end of the story is that all the sacrificing wasn't for nothing. Dramatic irony kept the reader and character from knowing it, but there was always going to be a reward for taking care of this penguin. Admiral Drake was testing Mr. Popper's passion for adventure and his respect for life, but ironically, that's because he's going to offer Popper what he's always wanted most: the adventure of a lifetime.
The ironic family
In a way, the penguins become a family for Mr. Popper. That signifies his quirkiness which drives a wedge between him and normal people, so they are an ironic family. But this ironic family reminds him of animal nature, and that heals him in a way over time, because he realizes the organic nature of his own life, and his own need for community and survival.
The irony of arrest
When Mr. Popper accidentally arrives to the wrong theater with his penguin troop, he gets arrested for ruining an event. The conductor has him arrested, which is condemnatory imagery. The arrest is an ironic reference to the fact that Mr. Popper is punished by communities of people as if he is evil, when he just made an honest mistake. He is trapped in other people's assumptions and prejudice.
The ironic promise of adventure
Mr. Popper is thrilled to be invited to the North Pole with Admiral Drake, but that is another ironic gift, because the adventure will take him through the Arctic tundra, and surely he will be exhausted, freezing, and perhaps in mortal danger in order to accept the quest. To him, though, that is salvation, because that is secretly the life he's always wanted, one with challenge and pain, but also adventure, wonder, and fulfillment. Still, it's no trip to the Bahamas.
