Without a mask, paintball can be very dangerous as paintballs travel at high enough velocities that they can seriously damage or even blind an eye. Just make sure you wear your mask at all times when you play paintball.
Other than eye injuries, which can be completely avoided with the use of a mask, paintball is as safe as any sport that requires running. I've found that it is actually safer than many sports because there is no physical contact between players in paintball, meaning that there's not even a risk of running into someone else.
Common paintball injuries include trips, sprains, bruises and welts. When you play outdoors there is the risk of stepping on uneven ground, tripping on roots and rocks and getting sunburned.
I've played lots of sports in my life and I would rate paintball as one of the safest. My most serious sports-related injuries have come from playing football, basketball and wrestling. My worst paintball injury came from me diving behind a bunker in the woods and smashing my elbow into a tree. It hurt, but it didn't put me out of the game.