Do we die when we inject air into viens using syringe?

Injecting air: Not a good idea - small amounts of air - what might be trapped in a syringe usually isnt a problem but large volumes - 20-30 cc can cause an air embolism and death.

Amount?: Remember that the veins are carrying oxygen-poor blood back to the lungs. Small amounts of air are quickly incorporated into the de-oxygenated hemoglobin which is "oxygen hungry". Air in the veins is generally well tolerated. It is not tolerated at all when injected into an artery. Even a tiny amount results in "air embolism" which paradoxically deprives served tissues of oxygen carried by blood.