If i leave a splinter in my finger will it eventually push itself out so i can get to it with tweezers?

Maybe: I would be cautious. A splinter can cause serious infeciton even if you think it is minor. If you cannot easily prick the splinter out with a sterile needle after a good prep of the area with soap and water and perhaps some alcohol, then you should seek help from someone qualified to assiste your. If a splinter "festers" it does come to the surface. The body will try to expel it.

Yes : In general most splinters that are superficial will work their way closer to the skin surface and can be removed by tweezer. Other ones that are in deeper will require a physician to remove them. Any sign of infection redness or pus should be properly examined by a physician for treatment.