I have seen this claim made by both EO and RCC apologists, tough they mean different things by it.
The RCC apologists say that the Church cannot err because the pope has a unique gift--the gift of not making any mistakes when he teaches the whole Church about a matter of faith, i.e. dogma, or morals. This unique gift was bestowed on St. Peter by Christ himself, and St. Peter's succssors possess the same gift. The EO apologists claim that while the Church may be in error here or there, as a whole the Church cannot err. By "as a whole" they mean over a period of time, perhaps centuries.
I believe both of these models are bunk.
Both groups are clever enough to limit the circumstances in which infallibility is operative. No one would dare claim a global infallibility in the Church.
The RC model initially may sound "cleaner", in that the infallibility is vested in a single office in specific circumstances. Not everything a pope says is considered binding on the faithful--only when he teaches the whole Church is he infallible.
The last part is the rub, though. You see, it is not possible to determine which parts of the pope's pronouncements are infallible and which are not by simply determining whether the pope is teaching the whole Church at this point or that one. Sometimes, different parts of a document or pronncement are infallible while others are not. A good example of this is the papal bull Unam Sanctam which one would at first blush assume fits the definition of an infallible statement. But that is not the case. Only parts, determined by RC scholars, are in fallible while others are not. So, I must depend on the fallible pronouncements of the Curia, along with my own fallible reason, to determine what the pope says. In other words, the infallible pronouncements are mediated through fallible people. So how can I have any confidence in what the fallible people say the supposed infallible pope says when he teaches the Church? It seems to me that the pope's infallibility is not useful in a practical way.
A similar thing happens with the Eastern Orthodox. Their claim is a little more nuanced in that for them it is possible for large parts of the Church to be in error, but "as a whole" and "over time" the Church will not err. This means that the Church could indeed be in error for a time, perhaps even a few hundred years, but the whole Church will never be in error; over time most of the Church will not be in error. Now, the EO add another nuance in that the Church consists of those Apostolic sees which are in fellowship with each other--only EO churches are the true Catholic Churches.
The problems with the EO approach are twofold.
First, their whole schema depends on assuming that they are indeed the original, the one Holy Catholic and Apostolic Church. They, and not the Oriental Orthodox, the Copts, the RC, the Lutherans etc. etc. are really the Church, and only within the EO communion can one be assured of God working among his people with the true Sacraments. This is simply assumed to be true by the EO apologists I have encountered. The problem is that the Copts, the RC, the Oriental Orthodox etc. each claim to be the true Apostolic Church--and they make similar appeals to history to "prove" it. This does not make the EO claims false, of course, it just means that unless one buys into the first premise--that they are really the Church--the rest of their claims are in doubt. I have discussed this particular issue several times with several capable EO apologists and I have yet to see them deal with this except to merely reassert the point, erhaps with the addition that I do not "understand" what they say.
The second issue is similar to the RC issue I outlined above. If the Church can take centuries to take care of error what guarantee do I have that what they say is apostolic today is actually apostolic? Also, how does one determine what "the Church as a whole" is? from my experience, it is simply defined as the Eastern Orthodox Churches, never mind any other communions who are not in communion with the EOC.
I enjoy these types of intellectual exercises, and I believe that the RC and EOC are trying to maintain their own "infallibility" by making so many qualifications that the original statement does not have any practical meaning.
We Lutherans, however, have a pretty workable definition of "The Church is infallible". We believe that the Church will never fail where the Gospel is preached. ("Gospel" here means forgiveness and nw life granted to the believer for the sake of Christ). The Gospel does not depend on any institutions of the Church to be without error, but on God's offer of forgiveness and victory over death he offers to us in the Sacraments of Holy Baptism and Holy Communion. The Church does not fail when she is faithful to her Lord, when she says what the Apostles said, when she repeats the blessed words of love the Father has toward us in Jesus Christ--not in any abstract notions of "infallibility" which are qualified beyond all recognition because of the manifest fallibility of the Church in history.
So it is true the Church cannot err, just not in the way that re-affirms the institutions of the Church. Someone said the Church is not an institution with sacraments, but a sacrament with institutions. The sacrament, or mystery of God saving us thriugh the Church will not fail, even if te institutions do.