Under ordinary circumstances an air bubble injected into a vein is harmless. It travels through larger & larger vessels until it reaches the right side of the heart, then gets pumped to the lungs where the small volume of oxygen and nitrogen are released in the process of gas exchange and exit the body.
A large volume of air, as @Lightlyseared noted, can be fatal because the heart just squeezes the air with no forward ejection of blood. It probably takes at least 30–50 ml in an average size person. Not your average air bubble.
A bubble in the arterial system, on the other hand, is a way bigger problem because it will travel through smaller and smaller vessels until it lodges in an arteriole where it can obstruct the flow of blood, causing the tissue served by the vessel to be deprived of oxygen. If, by chance, this occurs in the brain it causes a stroke; in the heart, a heart attack, etc. If it’s not in a critical organ (for instance, if you accidentally inject into the brachial artery instead of an antecubital vein in your arm) a single arterial bubble is still probably harmless.
The real problem is when there’s a pathway between venous and arterial circulations. About 10% of the population has a patent foramen ovale—a potential passageway between right & left atrium of the heart—that could act as a conduit to shunt a venous bubble to the arterial circulation. Normally, however, pressure gradients keep this from happening. With heart failure or other abnormal conditions, that might not be true.
Bottom line: Venous bubbles are probably harmless, but it’s best to avoid them.