Why do medical professionals tap syringes before injecting somebody? What happens if they don't?
It's so that the correct dosage of medication to be administered can be drawn in the syringe and to get rid of any air bubbles incidentally drawn.
The syringe has milliliter (and fractions thereof) markers on the barrel. In order to get the right volume of medication from the vial/ampule, one has to get rid of any air bubbles incidentally drawn in. By holding the syringe vertically and tapping it, you encourage bubbles to float to the top of syringe, then pushing the plunger expels any air bubbles and extra medication that might be in the syringe.
A tiny bit of air injected into a vein will not usually do any harm to a normal person. It merely gets dissolved in the blood or tissue it is injected into.
If you are interested in knowing more about it, you can read this Medscape article, which states:
...it has been estimated that more than 5 mL/kg of air displaced into the intravenous space is required for significant injury (shock or cardiac arrest) to occur. However, complications have been reported with as little as 20 mL of air.