Does a patient shot with a lead bullet need to have it removed to avoid development of lead toxicity?

Depends.

Bullets lodged in joints should be removed because lead toxicity commonly develops as the bullet is solubilized by the acidic synovial fluid.  Bullets lodged in soft tissue however generally don't cause a problem as they are often walled off and not exposed to body fluids.  Nonetheless, if a lead bullet can be easily extracted it probably should be as there are case reports - albeit rare - of lead toxicity developing from  bullets left in non-joint spaces.


Source

Najibi, S.  et al.  "Management of Gunshot Wounds to the Joints"  Techniques in Orthopedics.  2006.

Murdock, C. et al.  "Toxic Lead Levels Treated with 2,3-Dimercaptosuccinic Acid and Surgery"  Journal of Trauma-Injury Infection & Critical Care.  1999.