Most of the lines in that poem rhyme "imperfectly," or as "off rhymes," (see
http://www.kingsnews.org/articles/off-rhymes-and-true-rhymes) if at all, as is seen in the following first verse:
'Turning and turning in the widening gyre
The falcon cannot hear the falconer;
Things fall apart; the centre cannot hold;
Mere anarchy is loosed upon the world,
The blood-dimmed tide is loosed, and everywhere
The ceremony of innocence is drowned;
The best lack all conviction, while the worst
Are full of passionate intensity.'
http://www.potw.org/archive/potw351.html
As for their pronunciation, "falconer" is pronounced as one might expect, while "gyre" is pronounced just like the first syllables of "gyroscope" or "gyrate." See
http://www.tfd.com/falconer and
http://encyclopedia.tfd.com/gyre, which include audio pronunciations of the words (American and British versions). In my view, "gyre" and "falconer" do not rhyme, although they do share a final "r" sound.
