Jonathan, Laurel and Emily have commented on whether it’s possible to write (read) a poem without a layer of irony.
What I find odd is a tendency to conflate irony with self-awareness. Self-awareness may be a necessary condition for irony, but I can’t see that they are the same thing.
I’m English and middle class, so I live among people for whom, in everyday life, sincerity is often a faux pas. Irony is the default mode of conversation in social situations. But that doesn’t mean you can’t turn it off.