All my life I’ve heard it pronounced as berry-ul but lately I’ve been hearing it as burr-ee-ul.
Is this new or am I just now being exposed to it?
I pronounce it the same as you. ‘Burr-ee-ul’ sounds like a regionalism. Where have you been hearing that pronunciation lately?
Mrs Piper and I both grew up in Saskatchewan. I say “berryial”, she says “burrial”.
Podcasts. Mostly history and murder mysteries, both of which contain stories of burials.
I’d probably look into where the podcaster (or guest) who’s pronouncing it burr-ee-ul is from then.
I say and hear berrial, but on occasion I’ve heard burrial.
When someone dies are they then berried or burried? 
I’ve heard both pronunciations. From the mouths of New Orleans area locals, “berry-al” is typical and “burr-ial” is rarely heard.
I’ve always said burr ee al
I was raised by N philly parents in S Fla. picked up many of their “accents”. My folks are dead and buried now so can’t ask them to say it.
Or it could be my own weirdness.
“Burr-ee-al” sounds like something Cleveland Brown would say.
I’m not sure what that character’s accent or manner of speaking is supposed to represent, but I did find this on the Family Guy Wiki:
I know it’s not right, but Berry-uhl.
As young kids we pronounced our home town at the time, Danbury Ct. as “Dam-berry”.
Berry-uhl for me; never heard it the other way.
Reminds me of BE speakers who say Strawburry instead of Strawberry.
Both pronunciations are right per Merriam-Webster. And “berry-uhl” seems to be the majority pronunciation in the U.S.
On this side of the pond burial would be ‘berry-uhl’.
If it helps, the name Muriel would be pronounced ‘Mur-ee-ul.’
From the PNW in the US (Seattle area).
I say “berry-uhl” and have always heard it that way.
I’m from Arkansas, and I am a berry person. It could just be my family. Not sure but that’s how I say it.
Wiktionary has /ˈbɛɹɪəl/ listed for both Received Pronunciation and General American, and that’s how I say it.
I wonder how much of this is where it is pronounced how it spells, and how much of it is the merry/Murray merger, where /ɛr/ (err) and /ɜr/ (urr) are always pronounced the same way. I know my former pastor would say things that way when he would exaggerate his (native) Texan accent.
In other words, I wonder how how many people say “burr-y-ull” also pronounce berry as “burr-y”.
Minnesotan - I say berry-uhl.
I wonder if the bur-ee-ul people pronounce bury, buried, buries with the same “bur”.
Modern English “bury” was Old English “byrgan”, pronounced /byrjan/. That is, the vowel spelled as “u” in Modern English was pronounced as /y/, a vowel between /ɛ/ (like “Ben”) and /u/ (like “boon”). That vowel doesn’t exist in standard Modern English, but was moved to /ɛ/ or /u/. So the variant pronunciations of “bury” and related words goes back a long time.
I don’t recall if I’ve heard the burrial pronunciation IRL but in Meet the Parents, Blythe Danner’s character comments how the cat , Jinx, likes to “dig, squat and burry”. I don’t know if BD pronounces it that way herself or if she’s using the sort of regionalism you’d find in Oyster Bay, where the fictional family lives.