The irony of the human Martian
Technically, it's ironic that the humans treat Mike as a Martian, because (although he was born on Mars) he was only a child when he was taken back to the earth, and on Mars, he is probably considered an alien. So the irony is that when Mike returns to his own mother-planet (where his parents were born), he is treated as if he does not belong, as if he is an alien to his own planet.
The irony of magic
This novel's main character is a bit of a wizard. He has access to serious magic, and he uses it throughout the book. Most interestingly, he helps the earthlings develop their own magic. So ironically, Mike's magic is one of the most human things about him, since all the humans who learn his methods can do magic. It seems that Mike's magical powers are unique, but he shows that they are not unique, they are just technical.
The irony of martyrdom
When Plato wrote The Republic, he included an allegory about a cave which is about religious enlightenment. He ends that allegory by saying that truly enlightened people will be martyred if they try to share their enlightenment, because the people won't understand, but they will understand that their rules are being violated. That's the basic irony that shapes Mike's demise in this novel. His enlightened message about human happiness being the object of life is too offensive to the religious communities of the earth, so Mike is murdered. It's ironic because he was just trying to show them joy and happiness.
The irony of shame and freedom
By not caring what others think, Mike is free. He doesn't care what they think because his memories of life on Mars provide him with permanent perspective. The earthlings all believe they understand life fairly well, but on Mars, they were doing magic and stuff. He isn't fooled by earthling culture. That means that Mike is immune from the shame that dominates most people's lives. That doesn't mean he doesn't make mistakes and learn from them, but he doesn't try to do what people want or expect. He lives in a house, completely naked, with many wives, who are all also naked. He is ironically shameless.
The irony of the ending
The science fiction elements of the novel make it feel completely ironic when the ending of the book is so overtly religious and unfathomable. The book argues that Mike was killed, and then taken into heaven where he begins his immortal afterlife by being awarded the position of archangel. That's fairly ironic in any story, but in this one, it is downright confusing. It serves to elevate Mike's opinion to a religious level of authority by showing him as a martyred saint.
