In an ordinary differential expression test, where the null hypothesis is log(FC) = 0, I can expect that in the absence of differential expression, the distribution of p-values from the test will be approximately uniform. However, for threshold tests like limma::treat, edgeR::glmTreat, DESeq2::results with lfcThreshold != 0 and/or altHypothesis != "greaterAbs", or even something as simple as a series of ordinary one-sided t-tests, this is no longer the case, since the null hypothesis is not a lower-dimensional subspace of the full parameter space, and many genes will be contained in the interior of the null hypothesis and have p-values at or near 1. With these kinds of tests, is there any equivalent rule to the "null p-values are uniform" rule described above for tests against an more typical null hypothesis?

If you read the TREAT paper, you'll see that the p-value provided by the function is an upper bound on the "true" value. The latter (such as it is) cannot be obtained as the null hypothesis is a composite across the interval bounded by the positive and negative thresholds, so the true log-fold change under the null could be anywhere in between. The best that can be done is to compute a conservative p-value based on the worst-case scenarios where the log-fold change under the null is right at the threshold. As a result, the p-value distribution won't be uniform, but that's okay, because the aim is to control the type I error rate at or below the nominal threshold. So, when I assess a testing scheme, error control below the threshold would be what I would be looking at; the uniformity of the p-values is just a side effect (though obviously, the closer it is to controlling it at the threshold, the less conservative/more powerful the test is likely to be, so it's a nice property to have).

Ryan, I think you've pretty much answered your own question. The null distribution can't be uniform, in fact it isn't even uniquely specified, although it will always be left skewed with higher coverage around the larger p-values.

I'm not sure that a histogram of p-values is that helpful. Anyway, to see significant DE, you would be looking for a bump in the histogram around the left limit at p=0, just as you would for a conventional test.