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.