The reasons I object to this talk of modern-day apostles would include:
1. It's an ahistorical argument. It can be demonstrated that the word 'apostle' is used in the NT to describe several different sorts of people - but two millennia have passed since then. We can't just ignore the fact that the word 'apostle' has had, since the 2nd century at least, a technical meaning in the church. It refers to the twelve, plus Paul. newfrontiers seem to want to skip over all this history as if it never happened, as if we could ignore the history of the church and live in the book of Acts. We can't, and we're not meant to.
2. It's an anti-ecumenical argument. It has a tendency to place evangelicals who do not recognise any 'apostolic influence' outside the church. The church is, after all, apostolic! And if we define apostolic the way newfrontiers would like us to, all independents and presbyterians are outside the church.
3. It's an argument that undermines the uniqueness of the eye-witness generation. This is not Dave's argument about the sufficiency of Scripture, although it is similar. The church is apostolic because it listens to the apostolic witness - whether actually written by the apostles or not. This witness - the witness of those who saw his glory - stands as the foundation of the church. The foundation is in one sense part of the house, but in another sense is a separate thing, the necessary precondition of the house. So the apostles were in one sense the beginning of the church, and therefore a part of it, but in another sense they stood apart from it. That is important - it means the church is always faced by a standard that is in some sense external to it, namely the record of the apostolic witness in Scripture. A continuing apostolate is theologically disastrous, because it must tend to undermine the distinction between the apostles, who always speak to the church in Scripture, and the church, which always hears the apostles in Scripture.
4. It's an argument that dethrones Christ in his church. This is a more controversial point, because it cuts against bishops, presbyteries and the like as well as contemporary apostles. Christ rules his church by his word; God's people are entrusted to that word, not to successor-apostles. I am of the opinion that this means congregationalism - I have quoted Barth to this effect elsewhere.
5. I can't help thinking it must be a personally unhelpful label for people to bear. But I can't say much about that, with my limited experience of those involved.
6. It is hardly reformed thinking! That's not very important, except that when you stretch definitions this far I wonder whether they mean anything at all anymore...
Here endeth the controversy. Perhaps.