One question I got in an organic chemistry exam many years ago was why RNA is not as suitable as DNA for storing genetic information. Essentially why it is inherently less stable than DNA.
With a bit of organic chemistry knowledge you can determine how the additional OH in RNA allows for water to attack there and cleave the RNA strand. This is very dependent on pH, but RNA is inherently less stable than DNA.
But that's not actually the most important mechnanism here, but it is one that is inherent in the chemistry of the molecule, there is no way around it.
Cells need to regulate protein production up and down. If mRNA were to live eternally, there would be no way to reduce protein production. So cells obviously have a way to remove mRNA. What actually happens is that mRNA is permanently degraded by RNAses, so any mRNA that isn't replenished will be gone after a while and protein production will stop. This is a fundamental part of regulation of protein biosynthesis, and you can find that in every textbook.