First, there is the dialect feature known as a "Linking R" (which usually happens in dialects where people don't pronounce the "r" at the end of words by default).
In this video, you can hear JFK saying "eleven billion dollah" then a few sentences later, he says "eleven billion dollars", because the word is follwed by an "i". So, his accent drops a lot of "r" sounds by default, but inserts them back in if the following word begins with a vowel. https://youtu.be/bOGWTEgta_w?t=52
So there is a perfectly articulate, quite educated man, who is making "non-standard" usage that is quite correct in his own dialect.
Another example is where people would typically say the word "car" as "cah", for example. "Pahk the cah." However, if the next word begins in a vowel sound, these same people will suddenly pronounce the "r" at the end of the word. The same person would say "pahk the car in the yahd", because the word after "car" begins in a vowel sound.
There is also the "Intrusive R" where people add a "r" sound to words wholesale (instead of merely pronouncing an "r" that was already there, which they were dropping before). You often hear this if the following word also begins with a vowel. You can hear this when people say things like "idear" instead of "idea". Here's a British gent talking about this: https://www.youtube.com/watch?v=UpX8NZMxp9Q
This happens in certain parts of New England, Britain, and Australia. In the USA, while some in New England are laughing at Southerners for their exaggerated vowels, the Southerners are laughing at the New-Englanders for their crazy "r" sounds.
Note: I don't agree with the above poster's "speech therapy" comments at all and I'm assuming this question was asked in good faith (which can be a mistake)...
This kind of drop accent happens in all sorts of languages for what it's worth including Dutch (drop the 'N' at the end of things like Leiden and verb conjugation) as well as Spanish (Puerto Rican dialects) so
But for New England, I'd say it's mostly the lahng Ah that gets ya. (that's the long R). It's more common to drop whenever a word contains an "AR" sound. The classic example being: "Pahk the cah in Hahvahd Yahd"
But sometimes it's added:
Examples: Draw(r), Idea(r)
