Hi @macrobiome,
Sorry for the incredibly long answer! I hope it is helpful 
This is so interesting!
Are there samples in tumor, intestines and feces groups that are sampled from one individual?
If they are samples from the same subject that would probably violate permanova's assumption of independence. I say "probably" because it is debated whether 2 separate microbiomes located on the same person are independent, but if you expect for there to be a transfer of microbe between the microbiome than it is definitely dependent and permanova is not the correct tool for this data.
There might be a trend but I think that there is another factor that isn't explained by these microbiome sites that is causing similarities between your microbiome sites. If one subject was sampled for all of your groups, the tumor microbiome of that subject might be more similar to the same subjects fecal microbiome than the tumor microbiome is to other tumor microbiomes. This could explain what we are seeing from your plots where fecal seems to be more similar to tumor in some cases than tumor is to itself.
I think this possibly supports your hypothesis of transferring microbes because if there are microbes that are transferring between these microbiomes sites that would increase the microbiome's similarity to each other and you would expect that to happen within a subject. This reminds me of a paper about PDAC that found transfer from the gut microbiome to the tumor microbiome and is really interesting: Tumor Microbiome Diversity and Composition Influence Pancreatic Cancer Outcomes - PubMed. You might already be familiar with this paper but I thought I would share anyway!
Also a little shameless plug, I develop a qiime2 plugin called q2-FMT, which is currently in alpha release, that helps assess engraftment of microbes in the recipient after a fecal microbiota transplant. I think with some minor tweaks this could work to track translocation of bacteria between different sites.
Basically, you would have to assign one or more of the microbiomes sites as your "donor" and the others as the recipient. This could help you track what microbes are being transferred between your sites, and how similar those sites are to each other.
I go into alittle bit more detail here if that is of interest: Colonization from inoculum - #4 by gregcaporaso and I am more than happy to help more if this sounds of interest to you!
By looking at the differences between the Unweighted Unifrac and Weighted Unifrac results we can see that abundance is a driver in finding significant differences. Because Unweighted Unifrac is not significant and Weighted Unifrac is significant and because Weighted Unifrac factors in abundances while Unweighted Unifrac doesn't, we can likely point to abundance as a main factor as to why weighted is significant. If you want to understand this more I would look into differences between Weighted and Unweighted Unifrac, there are a lot of sources out there that can explain the differences more thoroughly.