(I don’t really want to continue the hijack about viruses being alive but …
A lot of living things need something else to reproduce, grow, etc. I have no idea why viruses are treated any differently than other life forms. Even humans would be unable to live without all the bacteria living in our gut and elsewhere.)
So, taking viruses as alive as anything else, back to the OP.
Immortal? In the normal life cycle upon entering a cell a virus isn’t going to last much longer. The host cell is going to die and the virus is likely to speed up things. Rupturing of the cell wall usually results which allows the new copies to disperse.
If it never enters a host cell, it’s a lot like a dormant seed. Now, plant seeds can last a very long time under the right conditions. A virus is less fickle. So, easily thousands of years if things are just right. But immortal as in “Till the Sun expands and vaporizes the surface of the Earth”, I don’t think so. DNA/RNA even on virus scale will break down eventually, no matter how carefully it is naturally preserved.
Now, if you sequence it, carve the genome is a long lasting rock, bury it deep on the Moon, etc. Then you can resurrect it a billion years from now. But that’s still not the same as living forever.