Suppose time is made of points, and suppose it's continuous. Suppose Sam suffers pain from noon to 1 pm and Sally suffers an equally intense pain from noon to 2 pm. Then Sam and Sally suffer the same number of equally painful points of time. So, it seems, we cannot say that Sally is worse off than Sam. But of course she is worse off than Sam. Hence we should either reject the continuity of time or reject the pointiness of time.
I think there is one gap in this argument. Even if time is a continuum made of points, it doesn't follow that our temporal experience is made of the same points. It could be that the basic perceptual units of time and pain are short intervals, let's say approximately ten milliseconds long (they may vary, too: sometimes time seems to be going faster, after all). And then Sally will have twice as many painful intervals, and the problem disappears. Maybe this works, but I think it's somewhat paradoxical. For this story to solve the problem, it seems that pain has to be suffered not at points of time, but at these short intervals of time. We cannot say Sam is suffering a pain exactly at 1:30 pm, it seems. In other words, we have something like Zeno's paradox of the arrow: at no time are Sam and Sally suffering, yet they are suffering.
But maybe one can respond in the same way that people have responded to the arrow. When we say that the arrow is moving at 1:30 pm, the truth of that statement is not grounded in what is happening exactly at 1:30 pm, but rather in the differences of arrow position between 1:30 pm and slightly earlier. Perhaps, then, we can say that Sam suffers at 1:30 pm, but his suffering at 1:30 pm is grounded not just in what happens at 1:30 pm, but in what happens over the basic perceptual interval that contains 1:30 pm?
Perhaps we can. But it's not quite so simple. For it is deeply implausible that Sam's being in pain at 1:30 pm is grounded in part in what happens after 1:30 pm. Yet a typical time t will be within one of the basic perceptual intervals of time, and hence some of that interval will come after t. So perhaps we should say that Sam's being in pain at 1:30 pm is grounded by the painfulness of the then-past part of the basic perceptual interval. Maybe 6 ms of the interval have passed, and those painful 6 ms is what makes Sam hurt. But then the basic perceptual interval of time isn't 10 ms, because it seems that a mere 6 ms of pain suffices (and if 6 ms, then by the same token 3 ms, and so on). So this is problematic.
A different move would be to say that although time is continuous, pain perception consists of discrete instants of pain. There are infinitely many instants of time between noon and 1 pm, but Sam only suffers at finitely many of them, and Sally has approximately twice as many instants to suffer at. My argument doesn't rule out this possibility, and it does indeed solve the problem. But it does it at the cost of positing a deceptive phenomenology. For a pain can feel temporally unbroken, and yet on this theory it occurs only at an infinitesimal fraction of the instants of time during an interval.
All in all, I think my basic argument and our experience of pain does provide some evidence against pointy continuous time. How much depends on how much we can rely on our phenomenology.