Hi caramel-
Allow me to expand on what tedabell said. You will always have some HPV DNA, just like your body maintains DNA of every virus you catch: chicken pox, staph infections, the common cold. That said, in the vast majority of cases, the human body will "clear" the virus. What this means is that the immune system suppresses the virus so much that a) it is no longer detectable, even by the most sensitive testing available, and b) you are no longer contagious.
This will take a few months at least, a few years at most. If you have one of the "low-risk" genital wart causing strains, then the average length of time for the body to clear the virus, even without treatment, is about nine months. For some people it is longer, for some, shorter. If you have one of the "high-risk" strains, it can take up to two years. This doesn't mean that low-risk or high-risk strains don't persist sometimes... they do, but rarely. If you do some looking into medical journals, you'll see that studies have shown that about 70% of people clear the HPV virus within one year. 90% or more within two.
So while you will always have the HPV DNA, you will very likely not be dealing with it your whole life. Just continue to go to your gyno, get your pap smears, and within a few months-two years, your paps will almost certainly be clear once more (provided you are not immunosuppressed with HIV or are undergoing cancer treatment).
Yes you will but the virus can be controlled by using medications such as aldara cream, condylox gel, podifilox when warts are present those medications can be prescribed by your doctor.