I have heard this interpretation before. My experience is that those who come up with this interpretation do so, not because of any evidence, but because they have a presupposition that all consumption of alcohol is sinful. In other words, those who I have seen using this interpretation are typically cherry picking proof passages to prove their doctrine rather than simply asking what the scripture means. My study tells me that there is no legitimate scholarly reason to conclude that the Greek word for wine used in the New Testament normally refers to unfermented juice.
Well, sorry for starting with a rather negative comment, as your question is a good one. The Bible definitely does NOT forbid the consumption of wine. There are several reasons we can be sure of this. First of all, Paul suggested to Timothy to drink a bit of wine in order to help with what was apparently a stomach problem (1 Timothy 5:23). The word translated as "wine" here is oinos, which means wine. It can be interpreted as unfermented grape juice but surely this is not the most reasonable translation for two reasons. 1. This is not the most common meaning and, as a rule, the most common meaning should be used unless the context demands otherwise, which it does not. and 2. Paul is giving "medicinal" advice. Unfermented grape juice has no medicinal value, while, fermented juice, in other words, wine, was then and even now is considered to have medicinal qualities. Drinking wine rather than unpurified water can prevent a great number of sicknesses of the digestive system because it kills bacteria. Unfermented grape juice does not have this property.
A second relevant passage is John 2:1-10. Jesus turned water into wine. Again, the word is Greek for wine. Every piece of evidence points toward this being wine, not unfermented grape juice. There is no possible way that the drink used at a wedding would be unfermented grape juice, rather than wine. In fact, we can safely assume that anyone interpreting this as unfermented grape juice certainly does so because of a preconceived idea that drinking alcohol is sinful, not because of good Bible interpretation. The person in charge of the wedding would not have praised teh wine if it was not fermented!
A third reason to assume that Jesus and Timothy and probably all or nearly all early Christians drank at least some wine is that unfermented grape juice only lasts as unfermented for a couple of days under conditions in the Near East in ancient times. Without pasteurization or refrigeration, unfermented grape juice will ferment very rapidly. Knowing that the church used the juice of grapes in the communion services on a weekly basis it is literally impossible for this to have been unfermented grape juice for the entire year. Even if you can argue that Paul was not referring to fermented drink in 1 Timothy, no one can legitimately claim that the church used unfermented grape juice.
A fourth reason to conclude that the Bible does not forbid the consumption of wine is that it does not forbid the consumption of wine. In other words, there is no passage of scripture which forbids the drinking of wine. There are many passages which teach against the drinking of much wine–of drunkenness, but none which outright forbid the consumption of wine. This can be said about both the Old and the New Testament.
A fifth reason I conclude that the drinking of a moderate amount of wine is that medical science tells us that the drinking of moderate amounts of wine (one glass or less per day) is not harmful and perhaps slightly helpful for health, especially for older people. I will admit that this is a rather weak argument and that the biblical arguments are stronger, but I thought I would mention it anyway.
I hope this helps.