I hear some people pronounce both 居る and 要る low-HIGH, and others I have heard pronouncing 居る as HIGH-low. Can you clarify whether these are correct, if the variations are regional or not, or if there is some other explanation for variation? お願いします！
In Standard Japanese both words are pronounced 「いる{LH}」.
In Nagoya which is where I currently reside, however, 「居る」("to be, to stay") is pronounced 「いる{HL}」 and 「要る」("to be necessary") is pronounced 「いる{LH}」.
So, your observation seems correct. Accents change regionally just like the colors of miso soup change regionally.
The words that are pronounced 「いる{HL}」 instead in Standard Japanese are 「炒る」("to roast"), 「射る」("to shoot an arrow"), 「鋳る」("to cast metal"), etc.
