Most names have what is called a Canon pronunciation like this one is in fact "EYE-REEN" but there are some who are given family preferred pronunciations like "eaR-ayNay". Like mine is "Stephen" which is canon as "STEV-en" however there are commoners that pronounce it as the spelling as pronunciation "STeF-ən".
Irene is unusual in that it has various pronunciations. In the U.S. I've only heard it pronouncd Eye-Reen, as with the famous 1930's movie star Irene Dunne. The British seem to prefer Eye-Ree-Nee as with the actress Irene Worth (although she was apparently born American). In other parts of Europe, the pronunciation ranges from Eh-Reh-Nee to Ir-Reh-Nuh.
I know an English-born lady in her 90s who's spent most of her life in Australia and she uses the 3-syllables. In one of the Leslie Dunkling name books there's a quote from a British lady called Irene - she explains that in the early 20th century an American musical was performed in London, and became a theatrical hit. It included a song about an Irene. But the American composer had set the word Irene to only two notes. The song caught on in England, and the 3-syllable pronunciation was doomed!