As I recall, there are notes on this in Mysteries, but here's my take on it.
Membership in the Order is a social thing, so that rather depends on the rest of the Order, neh?
Who's going to notice that some alchemist has become immortal, and not just really good at longevity rituals? He's in.
The living ghost? Would probably go out if the Order knew - communicate through letters and proxies and no-one will ever know.
Ascended Daimon? Same deal, but (IMAO) much harder to conceal.
Nope - but I have a Tytalus who'd love to Ascend. He doesn't know it yet though.
Mythical - as in no Hermetic Magus has managed yet, except there are those rumors about some of the founders...
I just pondered this deeply, and would offer a second perspective:
They'd all know Parma Magica - the Order would have to retain them as members, or hunt them down and kill them.
In the case of the Daimon, that's not going to be easy.
As for the Living Ghost, make sure you do this in the Rhine tribunal, where your rights and membership of the Order is protected by tradition
I think there are a lot of implicit assumptions being made in the original post and the replies that may or may not be reasonable. Just what is meant by immortal? I assume that would be an age thing. But maybe you're specifically including immunity to Warping, too? Here are the ways I remember grant immortality in some sense:
Ascendancy to the Hall of Heroes
Becoming (need not actually be completed, even for both immunities above)
Bonding (assuming you can make a lasting deal)
The Great Elixir
Guardian of Nature
The Path of the Body
The Path of Walking Backwards
other Transformed Being such as drowned men
Notice that a number of them do not required you to have given up your humanity.
Never seen a magi PC gain immortality, and I'd be hesitant to allow it without it being a goal achieved over a set of campaigns, rather than the outcome of one. That said, if the player can inspire good stories from it then I'd go there happily.
In two Ars games I did have an NPC who had a version of immortality - akin to trapped soul with a body maintained by magic. The two games were set in 1020 and 1197 Scotland and it was the same NPC present in both, which gave the players in 1197 a firm idea that the particular magus was powerful. In both settings the NPC was limited by Twilight. I feel the best PCs and NPCs need limitations, and Twilight is something which serves as a wonderful lever in that regard - the idea of having an immortal magus with no twilight ramifications breaks my ideas on setting. Such a Magus would rule the world.
I never played with immortal magi. I did start a saga that had them at the background, but it fell apart before the reveal (much before the reveal...). That said, IMO:
Misguided souls clinging to their earthly lives instead of to salvation.
Ascended characters too often engage in sinful encouragement of worship, not content in the place they have established for themselves in the Magical Realm God has constructed. They are thus prone to sin.
The Great Elixir at least lacks this sin-trap, but is still base clinging to the character's pitiful life.
Living Ghost is taking this to the extreme, literally clinging to life beyond death. It's pathetic, really.
Becoming means new faeries are created, which is poor waste of the character's soul; the poor soul will now never reach salvation, and is reduced to a mere shadow of life, lacking any vitality of its own.
Binding of course is Infernal, and thus represents complete rejection of the Divine and is inherently sinful.
Things like Guardian of the Forest aren't immortality at all - just longevity. You still accrue Warping.
I think they mostly just don't. They are immortals because they become creatures of one Realm, and thus lack powers from other Realms.
There is no official ruling, certainly not Order-wide, as they're so rare and unknown. Should the issue be raised - it depends on the type, and on the specifics of the case.
Alchemical immortality (The Greater Elixir) will probably keep you in the Order. Such immortals would be seen as "pure" and mystical, and would likely gather a great following.
Ascended hierarchs will get a mixed response. Some will think of them as gods and demand they won't be "demeaned" by removing them from the Order, while others will see them as Magical beings and as such no longer human and appropriate members of the Order. I suspect who will carry the day depends strongly on the circumstances, but favor scenarios where the spirit is respected - this is especially appropriate for the Rhine tribunal.
A Transformed Being such as a Great Beast will of course be officially respected as a member of the Order, under Bjornaer pressure. A Transformed Being such as a Drowned Man will face difficulties similar to the Ascended, but I think will be more easily accepted as still-a-member.
Finding a Living Ghost will send off a Quaesitorial investigation, and once it was clear that the character is truly dead he'd be out of the Order. However, in my saga above a Living Ghost tradition was secretly leading House Tremere and their members would have, thus, been accepted as "actually, they never died - not really" on pain of another Schism War.
Things like Guardian of Nature, Immortality of the Forest, Cult of Vesta, which aren't true immortality, will keep you in the Order, and may be admired.
For the above saga - not as rare as you'd think.
The Tremere/Disciples of the Worm had lots of Living Ghosts, perhaps half a dozen or so, including Tremere himself.
The Astrologers (another mystery cult, practicing Ascendacy) had a few Hermetic magi "stars", plus a small roster of ancient Pharaohs and Roman Emperors.
There was one Alchemical immortal touring Europe.
There was one, maybe two, Tytalus immortals using Binding still around, despite the purge following the Corruption.
And Merinita was still a Guardian of Nature.
I didn't plan on using them, but the setting supposedly included some Merinita magi who used Becoming too, probably including Quaendalon himself, some Criamon mystics, and Great Beasts.
All together - that's quite a lot of immortals crowding the setting, no?
True should have put more detail in what forms of immortality I mean. Primarily to me it true immortality would mean no other power can make them mortal and they no longer have to worry about warping. To be honest though any responce about any of the forms is welcome. I enjoy reading what all of you guys have to say just because most people here are interesting people.
This is one of the best ways
This is another flawless immortality. Unless the rumors mentioned about it is true.
Personality is altered and the infernalist still warps.
Requires the magus to keep a faerie alive forever keep the deal alive forever and if the bond breaks even once he auto ages to dust if he is too old.
To me this is the best way.
Been awhile since I read about this fairly sure it still causes warping though.
The magus won't age for sure but he will most likely get all the locals to get the pitch forks and torches out.
Barely immortality considering the magus is dead but it matches with immunity to warping.
Now this I forgot about it has more flaws then some become of the limitations the magus has to go though but has a lot of advanges as well.
TMRE (in the Immortal Magi appendix, I seem to recall) goes into some detail about this. In a nutshell, and this is a ruling consistent throughout the game line, a being with Might may not have Powers or Supernatural Virtues and Flaws associated with other Realms.
I see I knew about that ruling of course but duel Realms is a confusing thought to me. Though I do believe that in realms of magic it mentions that the abillity turns pure magic realm need to reread that to be certain.