Virus fragments can exert pathogenic effects inside a host, but that's distinct from being infectious.

To be infectious, a virus particle must be intact. This includes the virus genome and the virus particle containing all the structural components properly assembled.

Take HIV as an example: most virus particles made are not infectious. In fact, only ~1 in 1,000-20,000 are. That can be due to misfolded proteins in the particle, an incomplete/damaged genome, or mutations incurred during replication, among other things.

Individual parts of HIV can exert pathogenic effects inside an infected patient. The HIV protein Nef is transferred to uninfected/non-infectable cells, modifying their physiology & creating a niche favorable for pathogenesis. But thats =/= production of infectious particles.

So what about SARS-CoV-2? Well unlike HIV, SARS-CoV-2 is an acutely infecting virus. SARS-CoV-2 RNA fragments have been observed in patients long after infection, but those are not infectious.
So what are their roles? It's unclear at the moment.

Conflating virus fragments and infectious particles is not appropriate. Infectious virus must be intact and capable of producing functional proteins. SARS-CoV-2 RNA fragments may play a role in pathogenic effects or immune surveillance, but that's not the same as infectiousness.