From my old gradeschool science class, and later in high school biology, per the definition of a living organism, an organism must:
- have a mechanism to reproduce (as in a typical example of a strain/species must be able to replicate or reproduce; it doesn't necessarily mean that each member of the strain/species must be able to reproduce, but merely that the strain/species has a means for reproduction),
- and it must also undergo respiration.
Viruses check the first box. It's that second box where viruses fall short.
While other organisms metabolize (break down) "food" to release energy, carrying out respiration, and thus grow or repair themselves, viruses don't do any of that. Once a virus is fully formed inside a host cell, it does pretty much nothing from then on, until it either gets destroyed or finds a opportunity to inject its genetic material into another host cell. And that's it.
You might argue that other living organisms (particularly small, single-celled micro-organisms) can remain dormant for very long periods of time if the conditions are right (very low temperatures, for example). Yes, this is true, but they do respire once the temperature warms up or the conditions change, making it respiration possible.
But viruses don't ever respire, ever. Sure, there is a mechanism for them to inject their genetic material into a host cell, but that's it. That's the only action they ever undertake. When not actively injecting their material into a host cell, they either exist in a state of doing absolutely nothing or they don't exist at all (e.g., get torn apart by their environment). They don't eat, they don't breath. Besides that one, singular act of injecting their material into a host cell, they don't do anything at all.
The fact that they don't undergo respiration is why viruses are generally not considered "alive."