Can the preposition via be pronounced (ˈ)vaːr in English? The answer is yes, at least in non-rhotic accents.
As you know, the term via can be (ˈ)vaɪə and (ˈ)viːə in both RP and General American (GA). In LPD3 (p.875) we find that the former is definitely the established pronunciation at least in British English, and that it is preferred by 88% of the British people who responded to John Wells's questionnaire (that number rises to 92% if one also considers those respondents born before 1942).
The latest edition of CEPD only gives ˈvaɪ.ə for British English but acknowledges both ˈvaɪ.ə and ˈviː.ə for GA (p.531). ODP has ˈvʌɪə [=ˈvaɪə] for British English and both ˈvaɪə and ˈviə [=ˈviːə] for GA, though for the latter accent ˈviə is prioritised (p.1158). Finally, the Oxford BBC Guide to Pronunciation (2006) has ˈvʌɪə [=ˈvaɪə] but adds the comment “[c]ommonly also vee-uh, especially in phrases” (p.412), where “vee(-)” in bold type indicates main stress.
Both (ˈ)vaɪə and (ˈ)viːə can undergo smoothing and compression in British English. So for the latter we can get the smoothed variant ˈvɪ.ə and the smoothed and compressed vɪə. As far as the former goes, we can have the following: ˈvaɪ.ə → ˈva.ə → vaə → vaː (or vɑː). If we then take the monophthongal vaː and add the noun phrase a mask (ə ˈmɑːsk/ˈmæsk) to it, as in the sentence Oxygen was administered via a mask, what we are likely to end up with in British English is something like (ˈ)vaːr ə ˈmɑːsk/ˈmæsk, with the so-called "intrusive r" phenomenon extremely typical of non-rhotic accents.
And (ˈ)vaːr ə ˈmɑːsk is exactly what my Medical English pronunciation students heard in a listening test I gave them recently. The phrase, which at first sounded completely incomprehensible to them, only became clear when I gave my students the chance to look at the audioscript in the book. Not being aware of the possibility of r-intrusion in British English, it is not surprising that some of them also struggled when it came to understanding the expression ˈæsmrəˌtæk (asthma attack) a couple of minutes later into the audio clip I played.
These examples show us once again that teaching English pronunciation in EFL is absolutely vital and that failing to do so may mean depriving our students 'of the right' to understand English as it is spoken by native speakers.