Viruses, in general, are obligately pathogenic i.e. they don't usually co-exist (as commensals or symbionts) with our body cells. Their mechanism of survival involves usurping of the host cell machinery to produce their own proteins and genetic material. However, as pointed out in the comments, there are some viruses such as the GBV-C which can infect the host but seldom cause any disease. Moreover, this virus has been shown to have a positive effect on the innate immune response and inhibitory effect on HIV .
(Also, have a look at: Do beneficial viruses exist? If so, what examples are there?)
In our genome, here are some remnants of viruses that had infected our ancestors long back in past. A prominent example is that of retrotransposons. Retrotransposons are DNA sequences that constitute a significant part of our genome and they are structurally similar to the genome of retroviruses (like HIV). Retrotransposons had been considered junk DNA in the past but recent studies have shown that they do play some important roles in our body. For example, L1 retrotransposons cause somatic diversity (genetic diversity at the cellular level) of neurons and may possibly contribute to increasing the plasticity of our brain[2,3]. Retrotransposons can also regulate gene expression.
Other than that, there may be infectious viruses that are present in the body but are dormant. These may become active later (when the conditions are conducive) and cause the disease in that individual or spread to another person.
You can perhaps calculate the weight of these viral remnants (retrotransposons). It won't provide any great insight to anyone. You just have to calculate the percentage of genome these DNA sequences occupy and multiply it with molecular weight of the genomic DNA and total number of cells in the body. I'll leave that exercise to you.