In short, Yes. But there is a big BUT. This can only work if the wart remover that you are using happens to be multi-functional. Devices such as Aven Buzz Wart Remover Pen also remove mole, wart, as well as a skin tag. Please note that other solutions that are available in the market are mostly made to treat only the type of lesion that they're made for. If you happen to have a wart freezing kit, it should not be used on a mole unless the manufacturer recommends.
Another important thing to do is to make sure which type of mole do you have before proceeding to remove it. Different types of moles require different types of treatment. There's no such thing as one size fits all type of treatment. You should also identify the wart that you have, if you are planning to remove a wart too. If you are unable to identify the type of mole or wart, you should always consult your doctor before performing an at-home procedure.
Not all moles are cancerous, but there's still a chance. Which is why getting it checked by your doctor before removing it is the most crucial part. For instance, if you remove a mole that was cancerous with a random DIY home remedy, it may end up leaving the cancer cells behind and the situation can become way worse. While on the other side, if you were to get a surgical excision from the doctor, it could have saved the day for you.
Although mole removal is generally considered as a cosmetic procedure and insurance companies don't cover the doctor's bill. But the initial diagnosis should be covered if not the mole removal process. It is best to just leave the mole alone if it is not changing color, shape, or size (these are some warning signs). But if you still want to remove the mole for cosmetic reasons there plenty of home remedies that you can use. While there's countless home remedies out there, some of the common ones are apple cider vinegar and mole remover pen.