Are there any acceptable reasons for omitting data in a publication (i.e. "data not shown")?
I think it is perfectly reasonable to not "show" data. For example, consider a multivariate randomized control trial with two groups. If there are a large number of partially correlated dependent variables, the data are not suitable for graphical or tabular presentation. What we are potentially interested in is if there is a difference between the two groups. We can "see" that from a single sentence about a statistical test and a comment that the data are not shown (so people do not go looking to find the figure).

In a biology example, maybe you are counting the number of intact cells after two different treatments. There may be hundreds of slices that result in two numbers. What exactly do you want to see?