Faced this twice now. Similar situations:
First time was on a ride to my brother's camp ground way out in the middle of no where in southern West Virginia. Maiden voyage or nearly so for my son's new bike and at the camp ground we noticed the partly worn head of a Phillips screw inbedded into the center of his rear tire tread. I have my on-road repair kit and air pump with me. There are no bubbles when tested with soapy water and tire is hard. Should we remove the screw?
Second time, last night, in my garage at home. Plenty of tools and too cold and rainy for any fun rides anyway. Avon Venom R Valk/VTX tire on rear of my GL1800 being inspected by me. Tire has 13,000 miles and is getting close to wear bars (time to order the replacement). Notice a very small metalic glint recessed into an small irregular hole right dead center of the chicken strip. Looks like a medium sized staple or finish nail with no head. Tire is holding air, no leak with soapy water. Pull the metal object or ride it till the new tire arrives?
In the first case, we took a Phillips screw driver and simply removed the screw. It turned out to be fairly short and would not have penetrated until the head wore off and allowed the shank to creep inward. No leak, tire was sound for use.
In second case, I took my shop knife and used the blade to pry and dig and work at the object to get it to finally creep outward. When removed it looked like the shank of a rather short thumb tack whose head was worn off and the shank was migrating toward the center of the tire. It was only about 1/8 inch long and the tire has no leak with it removed.
I figure riding with such objects in the tires is just asking for a deflation at speed. Since I carry tools, air, and sticky rope plugs with glue I'd rather chance the flat upon removal than when under way. If you get caught out in the middle of no where'sville without tools or repair supplies; then I would not advise removal since the tire may well go flat and strand you; ride slowly and alertly to the closest source of help or supplies.