Non-rhotic dialects and intrusive r
Solution 1:
I can only offer this bit from the venerable alt.usage.english group:
Many nonrhotic speakers (including RP speakers, but excluding most nonrhotic speakers in the southern U.S.) use a "linking r": they don't pronounce "r" in "for" by itself /fO:/, but they do pronounce the first "r" in "for ever" /fO: 'rEv@/. Linking "r" differs from French liaison in that the former happens in any phonetically appropriate context, whereas the latter also needs the right syntactic context.
A further development of "linking r" is "intrusive r". Intrusive-r speakers, because the vowels in "law" (which they pronounce the same as "lore") and "idea" (which they pronounce to rhyme with "fear") are identical for them to vowels spelled with "r", intrude an r in such phrases as "law [r]and order" and "The idea [r]of it!" They do NOT intrude an [r] after vowels that are never spelled with an "r". Some people blanch at intrusive r, but most RP speakers now use it.
This actually quite clearly suggests that the linking R is not used by all non-rhotic speakers, expressly excluding "most nonrhotic speakers in the southern U.S.".
Solution 2:
This feature is called intrusive r, as others have pointed out. Bryan Gick, from Yale University, writes:
Intrusion typically refers to the presence of a non-historical consonant between two heterosyllabic vowels. ... All dialects having intrusive r also seem to require two subordinate processes: r-vocalisation (the reduction or apparent complete loss of some or all coda /r/'s also known as r-loss, r-deletion, r-dropping or simply r-lessness or non-rhoticity) and linking r (the non-deletion or reinsertion of these historical /r/'s when followed by a vowel-initial morpheme)
r-vocalisation. Historically, the first process that led to the present state of the intrusive r was the reduction or loss of coda /r/'s before tautosyllabic consonants. Jones (1989:ç5.3.3) gives orthoepic evidence of the dropping or reduction of /r/ in this environment from as early as the 15th century, citing such alternate spellings as: bersel/bessel, hors/hos, harsk/haske, morther/mother (for 'murder', not 'mother'), quart/quat, etc. By the 18th century in London the general reduction of coda /r/'s was a thoroughly well-entrenched, if not well-accepted, practice.
Linking r is a process common to many r-vocalising dialects, whereby a historically attested final r is not vocalised (or is vocalised and then reintroduced, depending on the account) when followed by a vowel-initial morpheme. This r is generally presumed to be retained or inserted either to serve as a `hiatus-breaking' element, or to provide a sufficient onset or coda to the following or preceding syllable, respectively.
Intrusive r is typically described as being identical in its surface manifestation to linking r, the only difference being that in the case of intrusion, no final r was historically present (e.g. idea > idea[r] is). r-intrusion only occurs in dialects having both r-vocalisation and r-linking.
Solution 3:
As far as I know, in casual speech, the intrusive 'r' is a feature of many non-rhotic English dialects, but I'm pretty sure that there's not any rhotic dialect that has it. Seems to be a hiatus repair strategy.
It occurs only in words ending with a non-high vowel when followed by a word beginning with a vocalic segment.
There's some work by Hartmann & Zerbian and Hock on intrusive 'r'.