Before we all rush off and set up a cyber institute for ecumenical textual criticism (see discussion here), we would do well to reflect on what can be best achieved by working alongside scholars from non-evangelical churches on a broad platform of some common understanding of Christian history and the biblical text. We need to ask who would be good collaborators and what it would be good to collaborate about.
If we're looking for scholarly contribution, I'd say that the groups most likely to have the calibre of scholar we would need would be the RC and the Greek Orthodox churches. Armenian Orthodox, being a group with scholarly tradition based within an independent country, might also be well resourced. However, on OT matters Orthodox groupings tend to be committed to a text quite deviant from the Hebrew, but which is not the 'Old Greek' of the 'LXX' either. So for the sake of argument we might confine co-operation to the NT. Obviously co-operation with the Greek Orthodox would be a natural thing for evangelicals committed to 'Byzantine-Priority' or Majority Text theories. We've already seen a common interest between certain groups of evangelicals and the Greek Orthodox with the production of The Orthodox Study Bible using the NKJV (let's not talk about the textual basis for that!).
What jobs could best be done with such wide co-operation? I suppose that one might be able to launch a major academic project taking apart some of the more cynical reconstructions of early Christians and their treatment of the text. One might also be able to co-ordinate response to books (or other media) that seek to discredit the integrity of Christian scripture (e.g. the likes of that code book by Dan someone or other). Anything else?