Listening to various vifax videos, they keep on saying Inis Oírr as iniseer, not inisheer. Surely it should be sh in that?

I'm not sure why, but it's definitely broad in this instance. Here's an island native saying it a bunch:
https://www.youtube.com/watch?v=_gZ8XMVORlY

Thank you. Maybe it is interpreted as Ini Soírr.

I suspect it's rather more likely comparable to the change in pronunciation of the copula depending on whether the following word begins with a broad or slender vowel or homorganic letter:

is sinne = ish-innh (in speech)

but

is mise = iss misha

That is to say, you may be right about the second word being interpreted as soírr, but the first word is probably still understood as inis, albeit with unstressed final s.

Ade, it seems you're right. The English placename is Inisheer though.

Anglicised placenames are a whole topic worth their own discussion. Sometimes they're useful for linguistic purposes, eg. "Sleaveen", "Slievegallane" and "Slievenamon" all demonstrate that the Munster pronunciation of Irish sliabh (and hence, sli- more generally) extended geographically to a particular region. Nevertheless, I wouldn't necessarily put too much stock in the English for the placename being Inisheer. All that means is that it was called that at some point by somebody. It certainly doesn't mean that the "iniseer" pronunciation is incorrect if that's what local native speakers are saying.

I have heard that the local pronunciation of Inis was /in´ıs/ rather than /in´ıs´/ on the islands themselves.

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WARNING: Intermediate speaker - await further opinions, corrections and adjustments before acting on my advice.
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), agus Gaeilge an Chaighdeáin Oifigiúil (CO).