The general theory is that a virus avoids evolving in a way that is too dangerous to its host because the sooner the host dies the less potential the virus has to spread. As the vaccinated have a weaker response to the virus there is no longer as strong evolutionary pressure against the virus becoming more lethal.

A real life/modern example of this can be found in Marek's disease in chickens. https://en.wikipedia.org/wiki/Marek%27s_disease#Prevention

The counter claim seems to be that because vaccination reduces the abundance of the virus there is a lower chance of mutations. It's hard to say which effect is stronger and it probably varies greatly by virus and by vaccine!

Disclaimer: I'm not an expert, just an armchair enthusiast.

I think it's too soon to say yet if the vaccines encourage mutation. But my understanding is quite a few of the significant variants come from places where there was significant vaccine testing: - Brazil (Gamma) - South Africa (Beta) - India (Delta) - Alpha (UK)

The causation may be backwards here. You need a decent enough medical system both to run clinical trials, and to detect variants.

Interesting theory though, I don't think it's true right now, but potentially there may be evidence I'm not aware of.