This is a great question...thank's for asking it! The ironic part about Dan Cody has to do with his origins versus his status in later life. Dan Cody's origins are very humble...he was not wealthy or from the upper class during his early life. Later on he makes his fortune through hard, hands on work mining for gold (or some such thing.) This provides him with an immense fortune.
So what's so ironic about that? I'm getting there...
Dan Cody is the one that James Gatz (aka Jay Gatsby) is employed by after a chance meeting at the lakeshore. Jay Gatsby is desperate to learn about how the upper class live and to adopt their ways. He sees the perfect mentor in Dan Cody, who seems to have everything (money, world travel, stuff) and soaks up his lifestyle like a sponge. The ironic part is that Dan Cody, unbeknownst to Gatsby, is a poor role model. He had none of the upper class origins that distinguish the "old money" people of the story. In short, he has the money to be in the upper class, but does not have the culture of the upper class.
You see, being upper class is more than just having a lot of money. At the time of "The Great Gatsby" (and still today, though to a lesser extent) in order to be considered "high class" you had to have good breeding and absorb the "unwritten rules" of the upper class. These are the things that Gatsby was never able to learn from Dan Cody, and they are the things that make Gatsby "stick out" as new money to the people like Tom.
To sum it up, Gatsby learns how to be "upper class" from a guy whose origins are distinctly "lower class."
That's my main answer.
As a little bit of a sub-answer I just want to add the idea in about Dan Cody's wife. From what we see in the story she is a bit of a "gold digger" and may have even been the cause of his death (it says that he died shortly after a visit from her.) In this way, Dan Cody's life is negatively impacted by her. Ironically, James Gatsby's downfall was over a woman as well. Just a little food for thought : )