Not trying to step on anyone's toes here, since this is a discussion I want to discuss, not have people rant or rave at me, this is just my opinion and I want to contribute to the conversation:
I think it would only happen if God let it happen, think Chronicles of Narnia.
Also, there are these universal laws in place that literally stop unclean things from being in the presence of God, so it would be impossible for people to reach heaven without having been cleansed first, thus the Atonement of Christ.
HOWEVER, lets not forget about the spirit world (the waiting period before judgment). What if they somehow found a way to enter the spirit world, no? And came back with data and whatnot. THAT would be cool!
I think that finding a physical "Heaven" out there in the many dimensions of the cosmos, the vast majority of society would become religious even more then it is today because there would be physical proof it existed.
However, upon finding it, then I think Humanity would realize finding our Creator would also be physically possible.
You're assuming heaven is a physical place that can be reached by technology. In this case, I would have to refer you to C.S.Lewis' "The Magician's Nephew" for my response.
If this were the case, even if they made it to "heaven", from what I gather, they wouldn't be able to get in, seeing as how it is a walled city. Angels I don't think would mind, considering God called us His children. And considering God has been taking people there for visits quite regularly, at least in my crowd of Christians, I don't think it would bother Him that much either.
Thus, my conclusion: only a Christian scientist would be able to complete such a feat. Say what you want about how paradoxical that is. They do exist.