In order to answer this, we need to define our terms to orient our thinking because the terms relate to and depend on each other. Space is limited so, briefly, irony is the quality in words or situations in which the expressed meaning is opposite of the felt meaning or in which the outcome of a situation is contrary to expectation.
- Verbal irony: You sit on my glasses; I say, "Thank you very much, that was helpful." What I feel and what I want known is opposite from what I say.
- Situational irony: A truck/lorry driver sells his car and walks everywhere. The professional driver's behavior is contrary to what is expected.
When examining The Waste Land for irony, it must be irony expressed in elements or themes, and the elements or themes must relate to the defining characteristics of modernism. It happens that irony itself is one of the defining features of modernism. Modernism was born in the aftermath of World War I in which industrialization showed itself in new, uglier light: industrialized warfare could decimate populations and ravage the entire planet. Bitter irony seemed to the modernist to be entrenched in the circumstance and in the state of the devastated world that survived progressive, industrialized modern warfare.
Eliot express this bitter irony in the very first lines of The Waste Land (the land that has been laid waste to):
April is the cruellest month, breeding
Lilacs out of the dead land, mixing
Memory and desire, stirring
Dull roots with spring rain.
Eliot alludes (another defining characteristic of modernism) to the torn and scarred land blasted by war that is now being begged by nature to produce lilacs, the early flowers of spring that represent innocence and first love.
Eliot calls the land "dead" and says personified April is the "cruellest month" because it is remembering and desiring past seasons of bountiful spring (ambiguous "remembering and desiring": ambiguity is another defining characteristic of modernism) and is cruel for shedding rain on the lilac's dulled roots that dying in a dead earth. In other words, the land is so dead from total warfare that even the lilac roots are dulled to the enlivening of spring that cruelly beckons anyway.
Therefore the irony in the opening lines provides a textual element that is relevant to modernism. This irony also succinctly conveys the modernist theme expressing the attempt to enter into universal consciousness. The lines explore the personified lilac's consciousness. Modernism sought to express the psychological consciousness of the collective world (to be attuned to Jungian universal consciousness), which encompasses all parts of the world from the dead land to flowers dulled and dead within that dead land to humans bitterly, ironically dead within themselves or laying dead in the waste land.