Leaving a splinter untreated is usually not advised as an infection can develop. Sometimes, the body can close off the split in the skin with the splinter still inside. Here, it can either cause no pain or be a constant irritant. It can develop infections depending upon how dirty the site of entry was. It is always recommended to remove the foreign particle from the body in order to avoid infections.
Keep in mind that no matter what method you use, it is important to clean the wound caused by the splinter with antiseptic soap and water before using any home remedy.