According to Healthline, injecting air into a vein can result in a potentially fatal condition known as an air embolism, which occurs when a bubble that is large enough to block the vein or artery enters the bloodstream. The bubble acts as a blockage, preventing blood from flowing to the heart, brain and lungs, which then results in organ failure if the blockage is left untreated.
According to Wikipedia, at least 100 milliliters of air need to be injected into the bloodstream to create an air embolism that is fatal. Death only occurs if the injected bubble is large enough to lodge in the heart, preventing it from pumping blood to the rest of the body. Most small bubbles of air injected into the venous system simply dissolve once they reach the lungs without doing any significant harm. Wikipedia also explains that these cases of non-fatal air embolism rarely cause any noticeable side effects.
Medical News Today explains that air embolisms are most common during medical procedures when doctors are injecting large quantities of fluid. WebMD explains that although possible, life-threatening air embolisms caused by injections are highly unlikely as long as the syringe is first checked for air bubbles.