Is it possible for a virus to live symbiotically with its host? Is the human body plagued with viral infections that do negligible harm, or even serve a beneficial role?
It is possible for viruses to live in mutualistic relationships with their hosts, these associations are often overlooked due to the devastating effect that many viruses can have.
To give an example in humans, when HIV-1-infected patients are also infected with hepatitis G virus, progression to AIDS is slowed significantly (Heringlake et al., 1998; Tillmann et al., 2001). Also hepatitis A infection can surpress hepatitis C infection (Deterding et al., 2006).
The table below, summarises some beneficial viruses across all organisms, and is taken from Roossinck (2011).
As mentioned in this question , Adeno-associated virus is often used for gene therapy. This is due largely to its predictability when injecting genes (1), however it is also used as it is not implicated in any human pathology. As it is a replication deficient/helper dependent virus, natural infection is much less likely.
The human immune system does mount an immune response against wild-type AAV, as Alexander Galkin discusses regarding an engineered AAV here. Despite this, I think that AAV does meet your criteria for a 'harmless virus' , even if it is unlikely to infect without human encouragement.
(1) Kotin RM, Siniscalco M, Samulski RJ, et al. (1 March 1990). "Site-specific integration by adeno-associated virus" . doi:10.1073/pnas.87.6.2211 .