What are some instances of dramatic irony, allusion, paradox and ambiguity in "The Things They Carried"?
Examples of these techniques are ubiquitous in The Things They Carried. What might help would be to narrow down to a particular story.
For example, "On the Rainy River" is a particularly strong source for such examples. In fact, the entire story is dramatic irony. As readers, we already know the main characters goes off to war in Vietnam, but through this story we get to experience the turmoil and heartache that went into his decision, all emphasized by the irony and paradox expressed in the story's last line: "I was coward, I went to war."
Likewise, the story is ripe with allusion and ambiguity as well. For allusion, you might look near the end of the story when the narrator imagines the throngs of people standing on each bank of the river. Many allusions to such patriotic figures as Abraham Lincoln, George Washington, etc are there.
Perhaps the most ambiguous element of the story is the behavior of Elroy. Does he fully understand the narrator's purpose in being there? Does he purposefully take him to within feet of Canadian soil and force him to make a decision? We--like the narrator--cannot be sure.