As used in the Beatles’ song in the OP, it would be very unusual to hear someone say that in the US. But I think most people would understand the meaning.
He paid dearly for his mistake. Yes, that wouldn’t be uncommon in the US.
The rent was too dear for us. I can’t imagine any American I know saying that.
Common in Australia
(Adelaide, Australia)
It’s more common for older folk here in Adelaide. My (maternal) grandparents say it, as do many people around their age.
Not so common in my mum’s generation and almost unheard of in my generation, though we understand that it means “expensive”.
'Cause that’s not really how it’s used. You’d probably use “high” for rent anyway, not “expensive.” (Though the apartment is expensive, the rent is high.)
I use dear and I hear others use it as well, but it’s usually used for slight comic effect. “Oh, the Manolo Blahniks were gorgeous, but I’m afraid they were just a bit too dear.”
New England; charmingly old-fashioned.
I use it, but never for something trendy & flash; good jewelry or a good coat might be ‘dear’, but this season’s hot shoes or bag or whatever is only expensive.
Both ‘beloved’ and ‘expensive’ mean ‘highly valued’.
I am familiar with that usage, but it’s something that I see more in print than something I hear said.
23, from Northern California.
My grandparents- all born in Ireland around 1909- used “dear” to mean expensive all the time.
But nobody may age ever used that expression in New York City.
I’ve lived in Pennsylvania and Louisiana, but the only person I know who uses it is my 103-year-old great-grandmother, a native of south-central PA.
The meaning of dear in this familiar excerpt from a speech means, I think, “highly valued.” That could mean either expensive or cherished.
"Is life so dear, or peace so sweet, as to be purchased at the price of chains and slavery? Forbid it, Almighty God! I know not what course others may take; but as for me, give me liberty or give me death! – Patrick Henry
I’m clearly showing my age here, because I use it fairly often.
Another Cannuck chiming in to say its not uncommon up here, but tends to be used by people 35 and older… as such its probabaly on the decline in usage.
regards
FML
Ditto. 27, from Arkansas
I grew up as the granddaughter of an English grandmother and daughter of an English mother – so “dear” in terms of prices was fairly common in the household. Same with most of those living around us here in Auckland.
But, the usage is giving way to “expensive” as the years pass. I agree that it seems to be a generational thing, tied in with British background.
You’re not old, just classical 
I have only heard it from my English aunt as she was talking to venders in Tijuana. I started to go and translate for her but they seemed to understand “too dear.”
I have never heard it from my lovely and talented mother who is from Geelong Australia, not from Ballarat as “Attack from the third dimension” would have you believe.
Coincidentally, we (3rd dimension and I) have the exact same mother.
Gee, what are the odds of that? 
Forty-something former New Yorker here, now living in the Boston area. I don’t think I have ever heard any person born in the US use the word “dear” as the Beatles did in the song in the OP.
The usage strikes me as quaint in a foreign sort of way. I certainly recognize the meaning but I read a lot of Wodehouse.
Very common in Ireland also.
I’d understand it, sure. But I wouldn’t use it.
I actually popped in to say that ‘When I’m 64’ was definitely not a Ringo tune - that one’s Paul all the way.
It had never occurred to me before that “dear”, meaning expensive, was not used throughout the English speaking world. It does seem to be older people who use it rather than young people - but everybody in the UK understands it, and nobody would comment or think it quaint if it was used in conversation.
As others have said, it is common enough that I would understand it, but not so common that I use it or hear it regularly.