Final R in non-rhotic accents

I know in non-rhotic accents the R is actually pronounced when a vowel follows. Example:

Order something (not pronounced in the last syllable of order)
Order at a restaurant (pronounced in the last syllable of order)

more bananas (not pronounced)
more apples (pronounced)

My question is: are there any dialects where those kinds of R's would not be pronounced anyway, even when followed by a vowel sound? How would it sound if someone didn't pronounce them and said things like "mo apples" (more apples), "fo eagles" (four eagles), "Mp3 playa on the table", "a snipa in the building" (sniper)... ? Thanks.

Sure. Non-rhotic US Southerners leave those final R's out, regardless of what follows, as do AAVE speakers.

Uriel makes a good observation - and in fact, I think some Southerners and AAVE speakers also elide medial /r/, so that, for example, "Carolina" would sound something like [kʰæ.əˈla:.nə].

I've heard that something similar occurs in South African English, although I'm not greatly familiar with that dialect.

This is true of certain UK dialects; in particular Tyneside and the most conservative form of RP

Thanks.
In the UK, however, I think it's much more common to add "additional" R's than it is to "remove" R's. What I am saying is that it seems much more common to say "Law and Order" as "Laura Norder" that it is to say "Four Apples" as "Fo Apples". So, it's not that common in the UK. Am I right in thinking so?

@Johnny: Correct. That phenomenon is known as intrusive R, and it's become predominant among non-rhotic British and Australians, and it's quite common in the northeastern US as well.

<<So, it's not that common in the UK. Am I right in thinking so?>>

The examples I mentioned would make up only a small minority and I'm not sure there are others. In particular, when I say "Conservative RP", I'm referring to a very narrow band of society.