Running smoke tests and verifying your work - absolutely, like verifying a portal page looks like you expect after a change, verifying a configuration setting is there. As far as running actual tests that involve creating test records (i.e. submitting a Catalog Item or transferring an HR Case), that will be a decision your organization needs to make. Personally, I do not like having test/dummy data in production and depending on your compliance policies, deleting it may cause some headaches during audits. This is why it's important to have good processes are in place around clones (ensuring at least one of your sub-prod instances is as close to production as possible), development, promotions, and testing in lower environments, so promotions mimic what go-live should be like. You should be able to find any errors, missing pieces from update sets, manual steps needed, etc.
