We don't know
I looked up in all of the published RPG books from Fantasy Flight Game that I had access to and found nothing neither confirming nor denying that:
- Emperor is a god, a basic principle for demon princes to exist, as they are aspects of a chaos god;
- Living Saints are the equivalent of demon princes of the Emperor;
- Living Saints are warp entities;
- Living Saints are demon hosts;
So, the answer is: we don't know. We don't know exactly how the Sisters of Battle faith works, it could be the Emperor granting them powers, or it could be powers granted by having great faith in something, which the warps translates into their power. Saint Celestine could actually be a saint, but she could also be a propaganda scapegoat from the Inquisition (PS: I find that unlikely, they are terrible at organizing themselves to accomplish something like that).
Quoting Chapter Approved:
"There are those amongst the Administratum who claim such things are simply mass hysteria brought about by the trauma of battle, while others maintain that it is a form of collective psychic projection. It is enough for the Ecclesiarchy simply to know that the Emperor marches alongside their armies and those who doubt His almighty power are fools of the worst kind. Those who have witnessed its effects simply know that faith can be a potent weapon indeed."
There is one distinction to make here though, which a lot of people are misguided about: What Saint Celestine is, and what the claimed Living Saints are, are two completely different things. Many of Saint Celestine powers have never been described as being used by other saints, especially her confirmed multiple resurrections, which is the thing that makes her most similar to demon princes, but also the ability to fly and survive in the void of space without any gear, the ability to resurrect her followers (Genevieve and Eleanor) and others (Guilliman), an ability also possessed by some demon princes.
There are a bunch of "saints" who are merely high members of the Ecclesiarchy, and not actually living saints. I will completely disregard those guys. As described under Saints on Dark Heresy: Blood of Martyrs (pg. 20):
Saints are those mortals who, by pronouncement of the Ecclesiarchy, have been elevated by the Emperor’s grace far above the bulk of humanity. They are exceptionally holy individuals, and their words and deeds are held to be the direct work of the Emperor himself. Throughout the ten thousand years of the Age of Imperium, millions of men and women have been recognised as saints, in many cases long after their death and only after decades of consideration. Most are known and venerated only within a specifi c region, but some, such as Sebastian Thor himself, are known across the length and breadth of the Imperium.
As opposed to the description of Living Saints from the same book:
Living Saints are those members of the Adepta Sororitas who are said to be so holy in mind, body and spirit that they somehow manifest something of the power of the Emperor himself. Living Saints are not beatified after their death, but rather are spontaneous manifestations of the Emperor’s grace.
However, the actual distinction to make here is that saints are declared so after their death, and living saints were declared saints while they are (or were) alive.
A short list of known living saints:
- Four of the six founders of the Adepta Sororitas (Dominica, Katherine, Silvana and Lucia);
- Saint Sabbat, of the Imperial Cult;
- Saint Drusus: the only male living saint. Note that he was "seemingly" resurrected, leading us to believe that he wasn't actually killed;
- Saint Mina, which just killed a lot of people mysteriously;
- Saint Arabella, with no known miracles and disappeared shortly after Saint Mina;
- Helena the Virtuous, Prioress of the Convent Sanctorum. Which the only thing we know about is that Saint Praxedes (whom "might still be alive") wore her armor;
- Missionary Joachim of Ulant, leader of an Imperial crusade against Abaddon's forces during the siege of Cadia. Declared a living saint after his death due to his deeds;
We also have Saint Anais from Dawn of War, which is a copy+paste in every aspect of Celestine under a different name, and we don't know how canon the game is.
Finally, we also have the Belle of Lost Souls (Dark Heresy: Enemies Beyond, pg. 18), which may or may not be a saint that was corrupted by chaos.
That said, Saint Celestine is the only actual saint that has the abilities that make her sound like "a demon prince of the Emperor", if that is actually a thing. Not only that, but we have several "saints" who never came back from the dead.
The list of RPG books searched is: Dark Heresy CRB, Dark Heresy: Enemies Beyond, Dark Heresy: Enemies Without, Dark Heresy: Inquisitors Handbook, Dark Heresy: Blood of Martyrs, Black Crusade CRB (why not), Deathwatch CRB, Only War: CRB. Of course, I might be missing a book that mentions the saints, but I tried to look at all possible books that mentions followers of the Emperor in a way or another. And of course, I also searched the fan-made wikis for references.
The book about Celestine, the Living Saint
They have recently published a novel about Celestine. I have not read this book, but looking at the synopsis of the book:
For each time she is slain, Saint Celestine finds herself trapped in a terrible realm of suffering and torment. There her faith in the God-Emperor will be put to the test so she may prove herself worthy of life once more.
I'm not sure what is this realm of suffering and torment, but seems like some kind of chaos dimension, maybe not related to the Emperor, or even any of the Chaos Gods, but certainly related to the Warp. It also sounds weirdly similar to how to orks get to manipulate reality due to their sheer faith that things will work the way they want them to work.
And a spoiler given by a random internet user:
'I do not doubt, for I have the Emperor to watch over me,' said Celestine.
'The cripple, the cadaver locked forever in gilded repose, the careless would-be-god for whose obscene ambitions all of mankind has suffered for ten thousand years,' hissed her reflection. 'That Emperor? He doesn't watch over you, Celestine my dear. He is little more than a ravenous corpse.'
Finally, quoting another guy from that same page:
If you’re specifically looking for something which fits in with an ongoing narrative then you probably won’t find that here. Instead, what you’ll find is an interesting character study wrapped up in an entertaining story, and a book which does an excellent job of reminding us that in 40k, even the brightest and most glorious characters are built atop dark and unsettling foundations.