For instance, I wonder whether roots *gʷʰér- "burn, heat" and *wer- "burn, heat" are related, as well as *gʷer- "mountain, height" and *wers- "mountain, height".
As Janus Bahs Jacquet mentions in the comments, there are a lot of instances within PIE where similar-looking roots have similar-looking meanings. But as Arnaud Fournet mentions, there's been no real success finding regular correspondences between these—for example, it doesn't seem that *gʰ- was any sort of regular prefix.
My view is, PIE certainly evolved from something, we just haven't been able to reconstruct that something with any certainty. And in many of the daughter languages, there are similar strange "coincidences" that make no sense until you have comparative evidence to work with. (For example, the alternation in vowels between Ancient Greek légō "speak" and lógos "word" doesn't follow any regular or productive pattern within Ancient Greek itself, but makes complete sense once you have enough information to reconstruct PIE ablaut grades.)
So I would speculate that many of these "coincidences" within PIE likely reflect remnants of some earlier pre-PIE system that we simply don't have enough information to reconstruct. And others actually are just pure coincidences. But unless we find a solid connection between Indo-European and some other documented language, there's no real way to know for sure; I've never seen an analysis that makes sense of these "coincidences" based purely on PIE itself.
Please note that *ghwer- "hot, heat" and *wer- "to burn" do not have exactly the same meaning.
That being said, assuming that *ghwer- and *wer- are more or less the "same" root would mean that *gh- is a prefix.
But it would seem that PIE did not have any prefix, so the most acceptable conclusion is that such pairs of roots are a chance coincidence.