Syllabic consonant /r/ is a very common sound in many accents of English, such as most American accents. However, syllabic /r/ is less common in British English and if it occurs somewhere then for sure there is an alternative pronunciation without the syllabic /r/. A very good word to exemplify what I have just said is the word “particular”. While an American is more likely to pronounce it /pər’tɪk yə’lər/, a British guy would probably say /pə’tɪk yə’lə /.
There are two situations where the pronunciation without syllabic consonant /r/ is acceptable.
The first situation is the case where more than one consonant precedes the weak syllable as in the word “history”. It can be pronounced with syllabic consonant /r/ /ˈhɪstri/ or without syllabic consonant /r/ /’hɪs’təri/.
The second situation is the case where only one consonant precedes the weak syllable as in the word “flattery”. It can be pronounced with syllabic consonant /r/ /’flætri/ or without the syllabic consonant /r/ /’flætəri/.
A particular example is the pair of words “Hungary” (the country in Eastern Europe) and “hungry” (the need for food). In these two words a difference in meaning depends on whether a particular /r/ is syllabic or not.
Hungary /’hʌŋgəri/ or very common in British English but not only /’hʌŋgri/ (with syllabic consonant /r/) Hungry /’hʌŋ’gri/ (without syllabic consonant /r/)
There are no cases where syllabic /r/ can not substitute either non-syllabic /r/ or /ər/.