schwa before /r/

Vowels before /r/ are often are altered in English. Because of this, pedagogical resources have a special term for these kind of vowels: "r-controlled" vowels.

There is a tradition for some accents of transcribing certain r-controlled vowels as diphthongs that end in a non-syllabic schwa sound. That is what /ʊə/ represents in the transcription /jʊərɵˈbɒrəs/. It is not two separate vowels or two separate syllables any more than the /aɪ/ in tie /taɪ/ is two separate vowels or two separate syllables. For clarity, it is possible to transcribe the non-syllabic element of a diphthong with an inverted breve below it (like this: /aı̯/, /ʊə̯/, /ɪə̯/) but this diacritic is usually omitted in English.

Since ɪə in this context represents only one syllable, it is theoretically distinct from a disyllabic sequence of vowels such as you say you have in seer. A disyllabic pronunciation of seer would typically be transcribed as /ˈsiər/ or /ˈsi.ər/ (the use of a period to split syllables is another thing that's optional in IPA). In practice, however, /ɪə̯/ and /i.ə/ are not always acoustically distinct.

It's not applicable to all vowels

Not all vowels have corresponding schwa-final diphthongs. The only schwa-final diphthongs that are commonly used in transcriptions are:

• ɪə, representing "r-controlled long e" (typically spelled with "e", "ee" or "ea" before "r")

• ʊə, representing "r-controlled oo/long u" (typically spelled with "oo", "u" or "eu" before "r")

• ɛə/eə, representing "r-controlled long a" (typically spelled with "a", "ai" or "ei" before "r");

You won't see transcriptions that use ʌə or ɑə.

In a usual Southern British English accent, "r-controlled long e" is a distinct sound from "short i". The symbol /ɪə/ is used only for "r-controlled long e". In this kind of British English accent, the word Sirius is pronounced /ˈsɪriəs/ (with "short i") not as /ˈsɪəriəs/. Because Southern British English speakers make a phonemic distinction between /ɪr/ and /ɪər/, we can say that the transcription /ˈsɪəriəs/ is incorrect for this accent.

American English speakers merge originally distinct vowels before r

But for most American English speakers, there is no such contrast: "short i" before r is merged with "r-controlled long e". This makes the choice between the transcriptions /ɪr/ and /ɪər/ no more than a matter of notation.

Different American English speakers have different intuitions about the identity of the merged vowel phoneme found in words like this. There might be patterns based on the region where the speaker is from, but I am not aware of research done on this.

There are also different distributions of certain mergers in American English. The most well-known variable merger involves the vowels found in Mary (r-controlled long a), marry ("short a" before r) and merry ("short e" before r). For many speakers, all three of these are merged as a single vowel phoneme that can be transcribed as /er/, /eər/, /ɛr/, or /ɛər/. Other speakers don't merge these vowels, or only merge two of them.

Wikipedia uses /ɪ/ instead of /ɪə/ in words like "Sirius" in order to indicate the correct pronunciation in British English

Wikipedia uses a "diaphonemic" transcription system, which means it combines transcriptions for different accents in order to provide more information to the reader. According to the rules of Wikipedia's key, /ɪə/ and /ʊə/ should only be used in words where British English has r-controlled long e.