It is extremely unlikely that these beliefs would affect Eich's job performance, though it is probable that holding those beliefs would affect the performance of Mozilla contributors - and for a group (corporation?) which relies on volunteers to function, this is an important consideration.
Plus, a big part of Mozilla's modus operandi is holding itself to a higher standard - and I don't see why that shouldn't apply to social issues as well, especially for those which Mozilla's community deems important. I have seen arguments that this is only a vocal minority making a fuss - I would imagine that the vast majority of those involved in Mozilla believe in equal rights for gay people, and many are upset (I am upset that he seems resolute in his views and refuses to have a conversation about it, for instance).
I do feel bad for the guy, though - it seems like he played a massive part in getting Mozilla to where it was today - but he partially brought this on himself, and not just by way of a $1000 donation 6 years ago, but also how he reacted to it recently.