If you have a normal pap does that mean you don't have the HPV virus anymore? Can the virus go away if you still have warts?

Not at all. HPV is a family of viruses. There are dozens of strains. At least 7 strains are associated with cervical cancer & 4 with warts(but not cancer). Once you have HPV it can remain in your system forever although the primary infection usually clears within 2 yrs. A negative Pap just means you have no signs of cervical cancer or pre-cancer at that time. It does not mean it won't be there next year.
HPV . Hpv or human papilloma virus is a family of related viruses that infect humans but cause different problems. Some hpv types can cause genital warts, but the hpv types that can cause genital warts are different from the types that can cause cancer. Hpv is the cause of 70% of cervical cancer globally. There are about 40 types of hpv that can infect the genitals or sex organs of men and women - the hpv types that are associated with changes on a woman’s cervix that can lead to cervical cancer include types 16 and 18, which are generally acknowledged to cause about 70% of cervical cancer cases. Just having the hpv itself is not the same as cervical cancer as many women have hpv. Few of them get cervical cancer if they follow their doctor’s advice for screening and/or treatment. Hpv can infect the normal cells on your cervix to turn abnormal. Over many years, abnormal cells can turn into cancer if they are not found and treated by your doctor. It can take 10 to 15 years (or more) for cells to change from normal to abnormal, and then into cancer. Abnormal cells are sometimes called “pre-cancer” because they are not normal, but they are not yet cancer. The pap test is a screening test for cervical cancer. It looks for abnormal cells on your cervix that could turn into cancer over time. That way, problems can be found and treated before they ever turn into cancer. Recently, a quadrivalent vaccine against hpv 6, 11, 16 and 18 has been shown to prevent infections and the cervical changes associated with them for those virus subtypes included in the vaccine. The effectiveness is nearly 100% in preventing new hpv 16 or 18 infections, which have been associated in more than 50% of cervical cancers. While there has been political controversy about the use of hpv vaccination before onset of sexual activity, the recommendations from the medical community support vaccination.