In a rhotic dialect of English (like General American English, GenAmE), the 'r' is always pronounced, but it is pronounced as part of the first word, not as the first sound of the starting word of the next sentence.
But you're probably thinking of a non-rhotic dialect, one that tends not to pronounce word final 'r's, like RP, the British English dialect likely to be taught. Non-rhotic dialects sometimes have linking-R's
As you point out, one might think that if the following word is in the following sentence one might still add in the linking-R.
It turns out that BrE does not do this, nor do any other varieties of English (there is too much 'space' between the two sentences.
So do not pronounce the R at the end of a sentence in BrE even if the following sentence starts with a vowel.