In the opening lines of W. B. Yeats' "The Second Coming" are the words "gyre" and "falconer" regarded as rhymes, and how to pronounce them?

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.