The tip of my right index finger has been feeling very cold lately. It's really annoying. Is there anything I can do about it? Should I get it checked?

It's unusual to have. The tip of a finger to have a cold sensation without affecting the other fingers or hand. If it persists or worsens, get checked for spasm of the blood vessel (raynaud's) , thyroid disease etc. Soaking it in warm water , quitting smoking (if u do) etc may help . Take care!
Possibly raynauds. Raynauds disease arterial spasm that affects fingers and toes. Affected areas often turn blue, feel cold and numb, and sense of touch is dulled. As circulation improves, the affected areas may turn red, throb, tingle or swell. Sometimes can affect one digit, not always same digit. Idiopathic raynauds is nuisance but not usually health issue unless secondary to connective tissue disorder.