Is it enough to discuss a higher mean in one set of results than another without statistical significance (as long as you are clear that significance wasn't tested)? Can this, in your experience, result in rejection by peer-reviewers?
Of course I can't speak for EVERY journal in the world. Who knows?!? But in principle, a difference in means, without further statistical analysis is 'meaningless'. You are applying a transformation that hides the variance of the underlying data, so it is impossible to interpret if the differences observed are likely due to a different mean of the samples, or due to simply random variation.
In addition, I would argue that if the journal does allow you to publish a difference in means without further analysis, you still shouldn't do it--and probably find a different journal to publish in.
It is a good idea to avoid meaningless "Null Ritual" statistical tests that give the appearance of statistical rigor, but are often deeply misleading. For instance, there is no point in performing a test for zero-difference between things that you know a-priori cannot reasonably be expected to be the same. For example, a test for zero difference in the proportion of journal paper abstracts that express an acceptance of anthropogenic climate change, versus the proportion when judged on the basis of the whole paper, tells you nothing whatsoever. A non-significant result only tells us that the sample is too small to reveal what we already know. The space in the abstract is too small to include a statement of whether anthropogenic climate change is happening, unless that is the specific purpose of the paper. However there is more room in the body of the paper, so it is more likely to contain such a statement in providing context for the findings. So we would expect the proportion to be somewhat higher in the latter sample. It certainly does not cast any doubt on the usefulness of the survey of abstracts (in case you are wondering, yes, this is a real example).
If the difference is too small to be of any practical significance, I'd argue that there is no point in performing a test for statistical significance either.
If you are going to claim that X is different to Y based on some data, then you need to perform a test of statistical significance. However, I'd argue that it doesn't necessarily need to have a significant result for the paper to be worth publishing, providing the conclusions are suitably circumspect (the method/reasoning may be of sufficient interest, even if the results are inconclusive). The main purpose of statistical significance tests is to enforce a degree of self-skepticism on the part of the researcher, and not much more than that. It is most useful when it give a non-signifiant result.
It depends on the situation.
If the mean of one result is obviously much higher than another one, then there is no need to provide statistical significance at least from my point of view.
If the mean is the same or very close to each other, then there is definitely a need to provide statistical significance.