The concern is that you will receive an air embolism. An air bubble blocks blood flow past it. The cells past it starve and die. In some locations, that can be critical. If the bubble is big enough, then it can go into your heart and impede your heart’s ability to circulate blood whatsoever. That being said, not just ‘any amount’ of air is necessarily lethal, and caution is taken to remove air from syringes (in part because having a big air bubble in your syringe means your measurement of dosage is off, and you don’t want to inject “eh close enough” dosages).
Courtesy of u/stuthulhu from [this thread](https://www.reddit.com/r/explainlikeimfive/comments/44ya5s/eli5_why_do_people_say_a_syringe_full_of_air_will/?utm_source=share&utm_medium=web2x&context=3)
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