How can you permanently get rid of a warts on hands? Using the over-the-counter liquid for warts, I seem to get rid of them for a few years. Then they grow back in the same exact locations on my fingers. Clearly, the virus is still present in the tissues

Many options. See a dermatologist. There are several options, though none of them are 100% effective. Freezing, chemical agents, injections, and laser treatments are some of the many options.
I . I think the key is persistence. You may just need to treat longer to completely eradicate the warts. It sounds like you are doing a great job with the home salicylic acid treatments, but i wonder if when you think they are "gone", there is still some left as you think. Why don't you try continuing the treatments for an extra month after you think the wart is gone? Also, you can see you doctor to evaluate and see if there are stronger treatments they can use to help completely eradicate the warts. With that being said, you may also be re-infecting yourself from wherever. Some people are especially susceptible to warts...You can get them from public pools, workout facilities, etc. So maybe you are just reinfecting yourself from the same source, and the weakest point in your skin's defenses are where the warts already were and is now scar tissue.