The most effective way to alleviate the pain caused by a wart is to have it removed by a doctor.
Having said that, there are some things you can do at home to try to alleviate at least some of the pain.
Corn pads are really useful for many warts. Basically, they're a tiny circular cushion with a hole in the centre that you place around the wart. Corn pads prevent the wart from being pressed down and therefore causing pain. This is particularly effective for warts on pressure points of the feet.
Painkillers
Over the counter painkillers such as paracetamol and ibuprofen may relieve some of the pain caused by warts. This is only a temporary measure, and as mentioned earlier, if you're suffering for a long period with wart pain you should seek the advice of a doctor.
DIY wart treatments
Doctors generally advise against using these, but some people do find that they work. You can buy specially designed salicylic acid based wart treatments that burn through the wart slowly over a number of weeks until it's removed. There are also DIY wart treatments that freeze the wart until it falls off.
Overall, the best way to treat a painful wart is to have it removed by a doctor. Over the counter treatments aren't always effective, and in some cases, they may cause complications.
Here's a video showing you how to use apple cider vinegar to remove warts: