Lee, if you're reading this, please correct me if I've misstated your position!
First of all, there are two things that everyone is willing to agree on. First, at all reasonable large distance scales, say above 10-42cm, the world is governed to an excellent approximation by relativity. In particular, the world appears to be smooth and continuous (a manifold, to be precise) and neither "space" nor "time" is entirely meaningful in its own right. Instead, our macroscopic world has three space directions and one time direction, and the natural symmetries and so on of the world treat these on an equal footing; we can talk about "rotating" between a space and a time direction. (In relativity, that sort of rotation is equivalent to changing one's velocity) So in daily life, space and time are really very similar.(*)
The second widely agreed thing is that, at some sufficiently short distance, the above description has got to be false. The basic reason is that quantum mechanics (QM) and general relativity (GR) don't play nicely together. QM says, for example, that the position of a particle is often not known precisely; instead we can say that it has such a probability of being here, and such-and-such a probability of being here.(*) General relativity, on the other hand (this is Einstein's theory of gravity) says that the presence of matter stretches space and time sort of like a bowling ball stretches a trampoline.(*) The problem is, if the object is at position number 1, over here, then space at position number 2 has been stretched, possibly even a lot - in fact, if the stretching is big enough, just what we mean by "position number 2" is murky. This means that even discussing what the probability that the particle is at position 2 is not well-posed; the shape of space is a function of the positions of matter, and the positions at which matter can even exist is a function of the shape of space. Something's got to give. In general, in any situation where both quantum mechanics and gravity are strongly active, the very notion of space as being smooth and continuous falls apart - much more so all the symmetries that connect space and time. This happens (among other times) at high energies, high temperatures (like in the early universe) and very short distances.
So now we come to the question of time. Basically, one of the major open questions in high-energy physics is what it is that replaces the notion of "spacetime" at these very short distances and high energies, and how our ordinary universe emerges from that at low temperatures. (i.e., if we have some sort of situation that looks nothing like space at high temperatures, how is it that when it cools down it ends up looking like spacetime, and not, say, like a jagged ball of fuzz?) There are two major approaches to this question right now.
Approach 1: When spacetime breaks down, all of it breaks down; the state of the universe at these high energies has no notion of any "coordinates" or smooth spaces or symmetries. As the system cools, this system coalesces into a smooth space and all of spacetime, and the symmetries that relate space and time, emerge together. String theory favors this approach.
Approach 2: The symmetry between space and time is, to some extent, an accident. At high temperatures, the notion of space breaks down, but the notion of time stays valid; imagine that space breaks into little shards or plaquettes, but these shards still have a smooth time evolution. As the universe cools down, these plaquettes coalesce to form space, and a nontrivial accidental symmetry emerges at low temperature that makes time - previously something very different from space - behave like just another dimension. Loop quantum gravity is based on this approach. (This is what Lee was referring to when he said "time does exist")
Although the second approach seems a bit strange in this context, it's important to realize that it has certain advantages: the equations are much easier to deal with (it turns out, for technical reasons, that the breakdown of a spacetime is a bit more subtle than the breakdown of just space) and there are situations elsewhere in physics where a dimension of spacetime emerges out of some quantity that at first looks nothing at all like a dimension.
My personal feeling about this matter, however, is that the former approach is more likely to be correct. The hypothesis that there's an additional, very deep, symmetry breaking phase, and that the notion of time is preserved to all energies and to all orders, does not seem consistent with what I know of the high-energy properties of physics. I believe for other reasons (entropy bounds, shape-of-the-universe constraints, some instincts based on what we do know about the breakdown of spacetime) that there must be a complete discretization of spacetime at sufficiently high energies, which could not allow a continuous time dimension to survive.
But ultimately, of course, these are just hunches - the only way to verify either of these conjectures is by experiment. Which is why I'll say for now "I think he's wrong, but I'm willing to be convinced."
(*) And there are obviously a lot of subjects scattered throughout this that probably require detailed discussion, but there's a size limit on both LJ comments and on the reasonable size of a digression on theoretical physics.
(Incidentally, my own research was on noncommutative geometry, and in particular with an eye towards how that may be a property of spacetime just beyond the point of this breakdown - it looks like spacetime starting to break into fuzzy patches that overlap and shift relative to one another, and those patches getting more and more disconnected as the energy goes up. My hunch is that this or something closely related to it is the first stage of the breakdown of spacetime, but that it breaks down even more at higher energies still.)
