When do I pronounce a non-existent “r” between adjacent vowel sounds?

Never!

It's called "the lazy 'R'", and to my Scottish ear it sounds terrible. However, some would say it's a matter of accent or dialect.

If you are learning English as second language, then you shouldn't use it at all.

It's not grammar, though, it's pronunciation.

Martin (and commenters) are over-stating their position a bit.

It's true that you never have to insert the R, but the idea that it's incorrect pronunciation stems from the idea that some English speaking accents are more correct than others. This is, of course, ridiculous.

It may be worth noting that John C Wells considered the intrusive R to be part of Received Pronunciation. However, since the minority of native English speakers speak with an RP accent, it doesn't really matter.

So, is it necessary to use the intrusive R? Only if you're learning English in an accent that uses it. If you're learning a rhotic dialect, then the intrusive R would definitely be out of place.

I support Dan Rumney's answer and I would like to explain a bit more.

In non-rhotic English accents —ones in which an 'R' sound is not pronounced if it occurs before a consonant or "prosodic break"— an R at the end of a word would not normally be pronounced, unless it was followed by a word starting with a vowel, for example in the expression "tuner amp". This is a linking R. Most English accents in England (including Received Pronunciation), Wales and the Southern Hemisphere are non-rhotic.

These accents also tend to insert an R in the same cases as above, for words ending with a vowel sound, even if an R is not written there, in order to avoid hiatus between the two vowel sounds. For example, in the phrase "bacteria in it", an unwritten R might be pronounced between "bacteria" and "in". This is an intrusive R. In your examples this happens even though "saw" ends in a 'W', because when spoken it ends with a /ɔː/ vowel sound, and is then followed by another vowel sound.

For a rhotic accent, there is no "Linking R" —the 'R' in those cases is already pronounced anyway— and a hiatus is preferred between two vowel sounds, instead of an intrusive R. Most English accents spoken in North America (including General American), Scotland and Ireland are rhotic.

To summarise when you may pronounce an Intrusive R:

If you are speaking in a non-rhotic accent and there are two subsequent vowel sounds with no "prosodic break" then you may wish to use the Intrusive R between them, to avoid hiatus.

The Wikipedia article about Linking & Intrusive R provides more examples and details about where and when these tend to be used. You might also be interested in the article about rhotic and non-rhotic accents.

British English developed a number of regional accents, in part from the history of who lived there originally, and what invaders conquered that area. You have the original Celtic influences in Scotland, Ireland, Wales, and Cornwall; Norse in much of eastern England; traces of Anglo-Saxon (Germanic) in the southeast, etc.

In the US, New England (Boston, Maine/DownEast accents in particular) was originally settled by people from Anglica, in eastern England. They tend to strongly exhibit the "R" behaviors discussed in this thread. In the Mid-Atlantic region, most of the early settlers were from Cornwall, Devon, and other southern areas, and they speak differently (not exhibiting the discussed "R" behaviors). This is the accent which has spread throughout much of the US as the "standard American" accent or pronunciation.

There are, of course, many other immigrant influences on US regional pronunciations (Dutch, Italian and Yiddish in the New York City area; French in northern New England and Louisiana; Swedish and Norwegian in the upper Midwest; German in Wisconsin and Pennsylvania; Spanish in the southwest). With widespread use of TV, radio, and sound recordings, the regional differences are dying out, replaced with a more uniform sound.