Dramatic irony involves the audience knowing something that the speaker/character doesn't. For example, in a horror movie when the audience knows the killer is outside and the actress runs out anyway because she doesn't know. This is NOT the type of irony in this poem, because the speaker and the audience know the same amount of information.
Situational irony is when something turns out the opposite of what you expected; for example, if I put a bucket of water over my doorway and try to trap my friend underneath it, but the water falls on my head instead. It could be argued that the last stanza of this poem is an example of situational irony, because the audience expects the speaker to finish the poem with more sincere words about her lover's gift to her, and instead we find out that she has been complaining all along that all she got was a rose.
Verbal irony is where someone says one thing but means another; often, sarcastic comments are also verbal irony. So the poem could also be an example of verbal irony, because in the first two stanzas the speaker is actually complaining about the rose when it seems like she is praising her lover for sending it.