Different Reactions to Seasickness
It is ironic that the stubborn and indomitable Lidenbrock is terribly seasick on the crossing while Axel is perfectly fine (46).
Fridriksson's Advice
It is ironic that Mr Fridriksson naively tells his guests that they ought to visit Snaefells; they are, of course, going deep into its bowels for their secret journey to the center of the Earth (53). Lidenbrock's amused responses add to the irony.
The Objective of the Journey
It is ironic that Lidenbrock never actually makes it to the center of the Earth, even though his plan all along is to do so. This irony can be found in the very title of the novel.
Axel's Daring
It is ironic that Axel, the person who was most worried about dying during the adventure (and most concerned that his uncle was leading the expedition down a perilous path) is the one who devises the plan to blow up the opening. This is the most dangerous, dramatic, and irrationally risky moment of the entire text.