@jayles: Thanks! Well, you're right - there are thousands of English dialects and accents all over the world! However, this "rule" I'm talking about has to be applied only to RP (Standard English RP) and non-rhotic accents of English. As David Crystal (1995: 262) says in his The Cambridge Encyclopedia of the English Language: "no other consonantal letter has such a variety of sounds, and is prone to such regional variation." My job is not to find a rule to fit them all, but only to understand whether for RP and non-rhotic accents my "rule" can be applied. :)

@jayles: So, If I am saying: "/r/ (the phoneme, i.e. the sound as in red) occurs only before a vowel phoneme (in British RP and non-rhotic accents of English) and that in every other case, it is silent.", is it enough to be considered as a "rule"? Thanks again!

Thank you very much for the post. May I ask you what happens with the "r" in the following words (in non-rhotic accents like RP): beware /bɪˈwɛə/, care /kɛə/, dare /dɛə/, there /ðɛə/, share /ʃɛə/, compare /kəmˈpɛə/, careful /ˈkɛəfʊl/, sphere /sfɪə/, figure /ˈfɪɡə/, and so on? In all of these cases the “r” is neither at the end of the word nor before consonant (rules that many BrE teachers teach for silent "r") – still, it is silent. Are there any rules that can be applied in these cases? What about: very, necessary, arbitrary, and so on - here the "r" is pronounced, but, even though in the middle of the word, there's no consonant before it (other rule BrE teachers teach for non-silent "r")? What's the rule here? What about the words: order, separate and the like? In "order", for example, the "r" is before a consonant - still, it is silent. On the other hand, in "separate" the "r" is in middle position, but there's no consonant before it - still, it is pronounced and therefore non-silent. What I am trying to learn is whether (or not) there are 2 separate rules for the “r”: one telling me when the “r” must be pronounced and one telling me when the “r” is silent. Am I missing something here? Thank you!
As I have spoken with other BrE experts, I would also like to ask you if the following conclusions are accurate enough and could be considered a rule for the pronunciation of the "r" sound (in British RP and non-rhotic accents of English):
1. "r" is silent in the following words: car, star, sister, mother, word, person, bird (/kɑː/, /stɑː/, /ˈsɪstə/, /ˈmʌðə/, /wɜːd/, /ˈpɜːsn/, /bɜːd/) because it is not followed by a vowel sound.
2. "r" is pronounced in the following words: read, write, red, Rome, grass, green, very, separate (/riːd/, /raɪt/, /rɛd/, /rəʊm/, /grɑːs/, /griːn/, /ˈvɛri/, /'sepərət/) and also in berry, carry, arrange (ˈ/bɛri/, /ˈkæri/, /əˈreɪnʤ/) because it is followed by a vowel sound.
Or, to sum up: /r/ (the phoneme, i.e. the sound as in red) occurs only before a vowel phoneme (in British RP and non-rhotic accents of English). In every other case, it is silent. Thank you!