Sticking some duct tape over warts or verrucae on the foot is something that you come across being recommended quite often. There are plenty of anecdotal comments that doing this does work, but the research evidence on duct tape for warts is not that good. There are several theories on how duct tape could work. While some of the theories might make sense, that still does not mean that it works better than a placebo.
If you want to try the duct tape, then how long should you apply it for?
There is no clear answer for this. Sometimes warts go in a matter of days or weeks. Sometimes it can take months. The wart could be ready to disappear on its own in a few days (they do that) and that is just the natural history of warts and it could have nothing to do with the duct tape. Other warts can last for months and be resistant to all sorts of treatments.
There is no harm in trying duct tape on your warts (unless you are allergic to the adhesive), but if you are having trouble with a wart, then it is probably better to see a podiatrist to get it treated properly.