The main problem is that in a timeless universe, there is no time, therefore there is no change, therefore no thing can become another thing, therefore the person at the beginning of the sentence can't become the person at the end of the sentence.

Maybe you have a timeless definition of person as a set of person-moments, and this allows you to nominally regard the person who started reading the sentence and the person who finished reading the sentence as parts of the same overall person.

Nonetheless, the denial that time exists implies a denial that change exists, and this introduces a disjunction between successive person-moments that is metaphysically incredibly radical because it is so utterly in opposition to experience. It denies that one moment becomes the next moment; instead they're just neighbors in Platonia, or something.

ETA: I've thought of another aspect of the strange implications of eliminating change from your model of reality. Not only is it an illusion to think that the present moment was previously a different one, it's also a mistake to think that the present moment ever ends. The next moment never arrives because that would require change.

And another way to express my point about the person at the start of the sentence being different from the person at the end of the sentence, is that, if change is not real, then each moment is a separate center of subjectivity, rather than stages in the evolution of a single center of subjectivity. By a center of subjectivity I mean something that is a locus and an agent of subjective activity, especially reflective or higher-order thought in which mental states themselves become objects of awareness. Most of us may lack the combination of "luminosity" and articulateness required to talk about these matters with great precision, but hopefully you will agree that thought sometimes has the characteristic of consciously compounding on itself: one thought becomes part of the next thought which becomes part of the thought after that.

I would also claim that part of this experience involves perceiving the preceding thought as something that you thought. Really, all I'm calling attention to here is one form of experience of time's passage, that is internal rather than external. Its significance in the present discussion is that, if there is no such thing as change, then again we have to treat significant components of the experience as illusory. If you are a static, timelessly-existing person-moment, thinking a higher-order thought which seems to have been created by reflective use of a thought you had just a moment ago - then you're deluded, because there was no "just a moment ago".

The only reason I'm not just flatly declaring that time obviously passes and that timeless ontologies are therefore obviously false, is the phenomenon of illusory continuity of consciousness after anesthesia. Roko reminded me of this once, and I think I may have experienced it myself, many years ago - the feeling that no time has lapsed, when in fact you just had an operation but you were unconscious.

The implications of timelessness for consciousness take this to the ultimate extreme: every moment of your life in which you feel as if the present moment evolved out of an immediately previous moment, has to involve the same illusion. As I have endeavored to point out, it's also a mistake to think that the present moment ever ends (we can't call this an illusion because you don't perceive the future coming, you only perceive the past going, and an illusion is a mistaken perception; so I'll call this a mistake instead), and aspects of reflective phenomenology have to be wrong too.

So even if I cannot truly disprove timelessness, I can at least point out that it is a skeptical hypothesis at least as extreme as "we're living in a simulation". That's skepticism in the philosophical sense - doubt as to whether there is an external world, or other people are conscious, or the world existed five minutes ago. Frankly, I think timelessness is more extreme than simulationism, because it asks me to deny something even more fundamental than belief in the external world, namely, belief that change is real and that the present that I experience flowed out of the past that I remember.

I have the strong impression that one reason people adopt timelessness (when they do) is that they only have timeless models. Modern physics spatializes time, and it seems it's always a small step from adopting a conceptual tool to believing that reality fits the shape of the tool perfectly, e.g. from "reality can be described by numbers" to "reality is numbers". Ever since relativity, time has become just another spatial coordinate, and experienced time is explained as something to do with the thermodynamic arrow of time, applied to memory formation and retrieval. Well, thermodynamics can create a particular timelike ordering of events in your space-time manifold, but I don't think it explains the experience of time, and it doesn't provide an ontology which "re-temporalizes" time, it doesn't give a genuinely time-like character back to time.

So hey wedrifid, I think your demure one-sentence response has given me a lot of what I wanted. I do maintain that utterly basic features of experience all but refute timelessness, and I suspect that believers in timelessness hang onto it only because they don't see a way to think rigorously about time in its phenomenological aspect, whereas they do have rigorous formalisms that don't contain time, except as a coordinate. So time, in the original and intimately known sense of the word, is sacrificed for the sake of belief in the ontological completeness of the available, timeless formalisms. (Incidentally, my own strong preference would be to find a quantitatively successful formalism for physics whose notion of time can be reinfused with the metaphysics implied by the experience of time, that is, time as change, and not just as another direction in space.)