Is it one or two?
One. Rhyme it with fire.
I’m pretty sure this is regional. For me, both “sire” and “fire” rather clearly have two syllables – sai-er and fai-er. Some dialects (esp. British ones?) are clearly one syllable though.
When you consider the rhyming words, it’s hard to make strong case for one syllable:
Buyer, crier, drier, friar, fryer, hire, higher, mire, pyre, tire, wire.
Of course, in the South that’s dealt with by changing the pronunciation to “far.” 
And if you’re a forest ranger you spend a fair amount of your time in a Far Tar.
Both fire and sire have two syllables for me. Where I live (Saskatchewan), film and milk are sometimes two syllable words as well.
When I was a little kid, first learning about syllables, I was told that words like “fire” and “sire” are correctly said to have one syllable, even though I always pronounced them as two.
I’m British, and I’d have thought two.
Rhyming with fire means it has TWO syllables. Only madmen and Brits pronounce /aɪ/ followed by /ɹ/ without /ə/ in between. And don’t get me started on people who only pronounce /ɹ/ if there’s a following vowel sound. That’s simple madness and probably why Atlantis was destroyed.
Two. Sy-er. Fr-yer. Fry-er. Cry-er. Py-er.
Etc cetera.
I bet you say “suh-weet” too…
These are not rhyming words for me. Higher has two syllables; hire has one.
Round here it sounds more like foire or foyer
I guess I’m going to have to give you that one. Said side by side there’s a difference, but unless you slow the typical speech down to hear it, you would be hard pressed to hear that second syllable in higher. Too many words are that way, IMHO.
This here.
The two adjacent vowel sounds do not create an additional syllable.
mmm
That’s a naive statement!
Well I’m a Brit — although we rarely call ourselves that — and some of us pronounce ‘Sir’ as ‘Sah !’ ( followed by clicked heels ); also I wouldn’t think many Americans would have used the word much as an honorific in the last 300 years.
Anyway, perhaps in various broad country accents — of which we had too many — some pronounced fire as fie-e-yer, but most now use the one syllable.
For an Italian sound, you may like to listen to this short film very hard, if you can find it.
The Sire of Vincigliata
Colour no less, 1913.
Sure, two adjacent vowels don’t necessarily create an additional syllable, but this is the rhyme -aɪə(ɹ). As wiktionary says
So… it’s dialectal.
I’m totally with Wiki on this one. Hire and higher are homonyms with two syllables, and rhyme with the two syllable fire and all the others on that page.