What Christian denominations/groups believe in biblical prophecies based on number of The Beast being 666?

Actually this is not a new discovery as Irenaeus (born 130 AD) argued that the number was 666 and was aware some manuscripts had 616 in error.

Anyway, in answer to how a different number might impact the various views that people take on Revelation I will try to summarize for you.

Historicist

From prior to AD 200 the 666 was assumed correct (versus 616) and as the Greeks commonly used numbers as code to represent hidden meaning based on numerical equivalency of letters, a tradition of 'latin people' was understood. In other words the Roman Government and during the reformation, who were almost all historicists, this became extended to apply to the papacy as well.

"There are here two things to be premised. First, the number given in the usual text is unquestionably correct, χξϛʹ; 666. For Irenæus in the most positive manner so states it, and also Hippolytus; nor, of the many solutions suggested by the Fathers, is there any one founded on any other number.—Secondly, there is a the highest probability of the language and number of the word meant being Greek, and not Hebrew...the famous name Λατεινος, given by Irenæus; itself “the name of a man,” viz. of the father of the Latin race, Latinus, and of which the numeral value is exactly 666. “(HORÆ APOCALYPTICÆ; or, A COMMENTARY ON THE APOCALYPSE, Elliot, p246)

Impact of 616 if it was not in error? I think someone would have simply found a different Greek word that was symbolic of Rome that added to 616 ;)

Preterist

As far as I know they almost unanimously understand the coding to be from Hebrew as symbolic of Nero. See here for an explanation of the coding of Hebrew letters (http://www.preteristcentral.com/666%20and%20Nero%20Caesar.html)

If our accepted manuscripts had 616, I think some other method would be used to identify a Roman power that persecuted the church soon after the letter of Revelation was written, as the preterist has the main thought that the letter is code to immediately direct and help the early church to prepare for imminent persecution by the Roman government.

Futurist

There are always many possible varieties to the futurist camp because as most everything has not happened yet, it could symbolize various possibilities potentially triggered by the imagination of any Christian leader watching the news. However, one popular idea is a number actually 'literally stamped' onto your hand or forehead to enable financial transactions, possibly electronically. Another idea is simply 666 being symbolic of man falling perfectly short from 777. In other words represents sinful man, and this starts to be similar to a 'symbolic view' accept it is expected to be understandable and recognizable when the event actually takes place in the future. So how 616 would affect things does not really matter as under a futuristic view, it would still be understood when it finally happened, whether related to digital commerce or not.

Symbolic

This view can easily explain anything because everything is only symbolic of timeless truths found everywhere else in scripture without really brining much new to the picture, besides an obvious 'holistic view of history' and the future in the light of those truths. Whether 616 or 666 it would have under this system the least impact. The way it commonly works under this view is to say if you show yourself to be a sinner by 'constant evil thoughts or words' (human nature stamped on the mind) or by 'actions' (sinful nature stamped on the hands)...or possibly 'obvious' sins, signified by everyone seeing your human nature on your forehead or 'secret' sins, less noticeably sketched on the hand. In any case 666 is representing human nature. If the correct number was 616 one would probably have to figure out how, or in what sense this is also symbolic of human nature. For example, 6 falling short of 7 plus one representing unity, plus falling short again, equals perfect depravity. I don't mean to ridicule anyone's method of interpretation, I just want to show how easy it is to develop a view that seems reasonable and yet obviously wrong (i.e. 616 is probably not correct, even though it could easily be used to seem correct).

Let's remember though, it is most likely the case that 666 is the correct number whatever it means exactly, as Irenaeus was already aware of the alternate texts and argued in strong favor of the traditionally accepted number 666.

Since 666 became the accepted number in Revelation 13:18, that is the number on which the various Christian denominations and groups base their interpretation of the prophecy.

Speaking for my own denomination, our primary theologian, Emanuel Swedenborg (1688-1772), in his interpretation of the number 666 departed from the most prevalent schools of thought among Christians, most of which applied it to some particular earthly ruler or empire, whether past or future.

He instead applied the book of Revelation as a whole, including chapter 13, to spiritual events, seeing them as symbolic and prophetic of corruptions in Christian belief, doctrine, and practice that came into existence in the Christian Church over the centuries. Specifically, he interpreted "the beast out of the sea" in Revelation 13:1-10 as referring to mistaken doctrinal views prevailing among the Protestant laity, and "the beast out of the earth" in Revelation 13:11-18 as referring to mistaken doctrinal views prevailing among the Protestant clergy.

This interpretation was not based on political considerations. Swedenborg himself grew up Lutheran, the son of a prominent Lutheran clergyman. So in these interpretations he was not criticizing "the other guy's" church, but the church he himself had grown up in. Also, he was not particularly attacking the laity or the clergy itself, but rather beliefs they held that he viewed as mistaken and unbiblical.

About the number 666 specifically, he viewed it as having both a good meaning and an evil meaning. In the good meaning, it would refer to everything good and true in the Bible. But since in Revelation 13:18 it is associated with an evil figure (the beast out of the earth), it there has an evil meaning. That meaning, he says, is the falsfication of everything good and true in the Bible.

If you're a glutton for punishment and want to read the original, rather technical version of his interpretation of the number 666 (translated from Latin into English), you can find it here.