There are two potential problems, as I understand it. First, learning a language, particularly an ancient language, involves learning ancient words for things that have no analog in modern life, whereas translation involves mapping those very things onto their closest equivalent in a modern language. It would be too easy using the interlinear method to start thinking that the Greek word 'krater' means a bowl or cup, when really it is a specific type of bowl used for mixing wine. That can really only be learned by consulting a dictionary, and probably a few pictures.
The second issue is only really a problem for people who are particularly familiar with the text in translation - it's very very easy to start mentally reciting the translation you're familiar with instead of actually parsing the original text. This is one reason that Attic Greek is preferred to Koine (New Testament), even though Koine is generally simpler. Even a secular person usually has enough familiarity with the Bible to make this an issue.
I certainly wouldn't say that this method can't be used profitably, but rather that one should be careful not to fall into those particular traps. The same criticisms could be made to relying on the Loeb editions.
Also, minor nitpick: The New Testament isn't really written in Classical Greek, it's in Koine. Going from Koine to any classical dialect is going to be much more difficult than going from any classical dialect to Koine, due to the simplified grammar and vocabulary of Koine Greek.