Evolution only requires "descent with modification". DNA is a mechanism for this, but it is not necessarily the only one.
> Prion replication is subject to epimutation and natural selection just as for other forms of replication, and their structure varies slightly between species.[19]
[19] is https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2848070
My impression is that DNA / RNA-based organisms have a high tolerance for changes to their DNA / RNA, such that many genetic changes won't cause them to lose their self-replicating ability. This is great for evolution.
Prions, on the other hand, seem like they would have a low reproductive tolerance for changes to their structure.