People believe that foreign objects in deep puncture wounds should be left in until medical care is available.
Is that belief true?
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With impalement, the penetrating object remains within the body [...] Such objects should not be removed in the field because they sometimes tamponade injuries to vascular structures and thus prevent exsanguination.
Source: "Penetrating Abdominal Injuries", S.J.A. Sclafani and S. Sheeran, page 382. Chapter 17 of Imaging and Intervention in Abdominal Trauma, ed. R.F. Dondelinger, Springer, 2004. Google Books
Some objects which are not large enough to produce effective compression of blood vessels should also be removed in controlled circumstances.
The most complete and detailed account of arrow wounds and treatments is Dr. Joseph Howland Bill’s “Notes on Arrow Wounds,” which is considered the “definitive work on American arrow wounds.” 
If the shaft was left in place, Dr. Bill’s treatment was to make an incision to enlarge the entry wound and slide a finger down the shaft to feel the depth of the wound and determine if the arrowhead is lodged in bone. Without the shaft in place the doctor was forced to search for the arrow by making a larger incision, probing through tissue, causing more trauma, and taking more time. It was much easier for the doctor and patient if the shaft was left intact until a doctor could remove the head and shaft as one piece. Further, there was always the danger that the arrowhead could not be found leaving the “angular and jagged head… buried in bone to kill – for so it surely will.” If, however, the arrowhead is removed properly, the wound was likely to heal naturally.
Not only is there a fear of losing part of the penetrating object, but barbed objects might also cause more damage as they are removed. In this report of a spear gun penetrating the skull of a 9 year old child, they advise that 
barbed objects should not be removed by retracing the route of entry
as that causes maximal tissue damage.
Large shrapnel pieces have the same issue as broken off arrow heads, difficult to locate deep inside tissues, and jagged edges causing even more trauma when removed.