Could a virus produce prions, leading to prion disease?
Perhaps not prions themselves, but viruses may possibly contain "pieces" or domains of proteins with prion-like chemical activity, which can catalyse refolding of prions from normal to disease state.

Mutation rates put limits on the size of RNA viruses. There is a higher cost for the average RNA virus to allot precious genomic space on encoding a normal protein that would take decades to be pathogenic, and which does not seem to help make copies of the virus.

Because DNA is a more stable molecule, and DNA polymerase has a higher fidelity rate than RNA polymerase, DNA viruses can have a larger genome. They could more easily carry code for encoding prion-like domains inside proteins.

There's a recent paper from George and Victor Tetz that looks for these so-called "prionogenic domains" (regions which encode portions of another protein that could potentially have prion-like behavior) in a catalog of viral genomes, and they find approximately three times more such domains in DNA viruses than in RNA viruses:


  We determined that more PrDs can be found in the DNA-viruses compared with their numbers in the RNA-viruses, and in the enveloped viruses, compared with that in the non-enveloped ones. This may be partially explained by the larger genome size and protein numbers in DNA-viruses.


While the domains themselves are usually part of proteins that are not themselves prions, infection could lead to expression of those proteins, which in turn provides the domain templates that invoke misfolding in the host of normal prions into pathogenic (disease-causing) prions:


  Some viruses may be implicated in the prion misfolding in humans since it was observed that the de novo appearance of prions can be facilitated by another PrD-containing protein


Also! There are retroviruses that have integrated themselves in our genome over millions of years. These are called "endogenous retroviruses" or ERVs, and have been called "junk DNA" by some biologists. However, some get expressed along with our normal genes:


  Although a large percentage of ERVs are replication-defective or are suppressed by host defense mechanisms, some ERVs can be expressed and replicated. The activation of an ERV gene can bring positive co-opted functions to the host, which may be evolutionarily maintained [13,20]. For instance, some ERV proteins may provide important functions during normal development; for example, human protein syncytine-1, which is an ERV Env protein, is involved in the formation of the syncytial layer in the placenta [30,31]. Moreover, the expression of Env proteins encoded by ERVs has been suggested to mediate host resistance to exogenous pathogens [32,33]. However, because of their analogy to exogenous retroviruses, the activation of certain ERVs has been frequently implicated in disease.


The authors observed that some people who develop Creutzfeldt–Jakob disease (CJD) have ERVs that get overexpressed:


  To evaluate the possible relationship between HERVs and human prion disease, we examined the retroviral sequences in cerebrospinal fluid (CSF) obtained from individuals with sporadic CJD. The frequencies of several HERV families, including HERV-W, HERV-L, FRD and ERV-9, were significantly increased in the CSF of individuals with sporadic CJD compared to the frequencies observed in normal control CSF. In addition, when compared to individuals with other neurodegenerative diseases that exhibit similar symptoms to CJD, such as dementia, the incidence rate of HERV-W and HERV-L were significantly higher in the CSF of sporadic CJD patients. Moreover, the frequency of increased HERV-W and HERV-L in the same samples was much higher in sporadic CJD than in either normal or other neurodegenerative diseases CSF samples, and there was no correlation between individual parameters, such as sex and age [61].


The cause may not be expression of proteins with PrD, but with the triggering of signaling pathways that lead to inflammation of tissues, which sets up conditions for normal prions to malform into disease-causing prions:


  According to these studies, ERV proteins may be closely implicated in the pathogenesis of inflammatory diseases, such as multiple sclerosis and rheumatoid arthritis (RA) [51,54,68]. In prion diseases, a correlation between the inflammatory process and neurodegeneration has been suggested because of the activation of large numbers of glial cells and the up-regulated expression of proinflammatory cytokines, which are pathological features of prion diseases [69,70].


Life is complicated.