Fingertip pushups are great for strengthening like little sprite mentioned, but also true that using correct form is crucial. When done right, they dont hurt, pretty good guideline to follow. The tip of the fingers are what is supporting you, with wrist straight. Dont ever let your hands slide down, at that point you a) put finger in their weakest position b)wrist is in its weakest position.
Aswell as its primary benefits, they let you do chest pushups with wrists alligned correctly, like with knuckle pushups.
I used to do lots of fingertip press ups back in my karate days. Several of my teachers talked of arthritis in later life from heavy hand conditioning.
I would only use techniques like this every day if considering a career in martial arts. Once or twice a week should be fine though.
Finger pushups are not the main culprit in this, alot of martial artists strengthen their hand by the break/heal principle, this has a massive risk of messing you up later on. If done correctly, fingerpushups are not dangerous, as they work with joints that are locked in the strongest position.
Does the strengthening of tendons and ligaments carryover to sport?
They are heavily used in Judo,Akido and probably other sports where grip is important. Strength in the hand comes from forearms muscles yes, but dont forget the tendons and pulley in your hand, they get tougher and more resiliant aswell.