I just saw a video of some beachgoers who rescued a bunch of dolphins that came in too close to shore and got caught in the shallows on a beach. But these well-meaning folks were grabbing them by their tailfins and dragging them back into the water. Now I understand they make convenient handles and I guess it’s better than dying, but can’t that injure the dolphins? Couldn’t it damage their spine? The body tapers down at the tail and mechanics of swimming wouldn’t require that the tail would have strength against longitudinal stress. I have no clue about dolphin anatomy but it hurt to look.
Seeing some of the acrobatics that dolphins regularly perform, I don’t think being dragged by the tail for short distances pose much of an issue.
It’s a nono, not just because of dolphin safety, but because the dolphin can throw a poweful kick to the humans. 
Most dolphin rescue sites recommend not attempting to rescue stranded dolphins yourself, but to call for expert assistance.
In any case, it is not recommended to use the tail, flippers, or fin as handles. In particular, the powerful tail can give a dangerous blow.
Off hand, since the tail is subject to very strong forces and has powerful tendons I wouldn’t think that pulling on it would do damage to the tail itself. However, dolphin skin is delicate and you could probably do some damage to it.
It’s mainly a risk management question… sure there’s a small risk of doing some superficial damage mainly to the surface tissues, but like you said a lot more damage would be done by stranding out of water. With the strong tendons and other connective tissue in the area, I doubt you’ll do much if any structural damage to the spine… and being right at the end of the spine I don’t think the damage you might do would matter much.
If a rescuer couldn’t get an ideal grip on my body so as to minimalize damage while pulling me out of burning car wreckage, I’d gladly have them yank me out by one arm or my hair or whatever. As long as they do less damage than what’s going to happen if they do nothing. My brother had a tooth busted during an emergency intubation a while back… sucks to loose a front tooth but he would have been dead in a few minutes if they didn’t do what they did.
Now that I can post better:
No, it is a no no, the tail is powerful and can damage the handlers themselves. Also dragging the dolphin (who is supposed to be able to swim immediately afterward and get back to the wild) may damage the skin and other flippers. This is not analogous to the car rescue situation. Yea, if someone needs to pull at my hand to get me out of a dangerous place, and dislocates something to get me out of the car alive, so be it. But nobody will expect me to be reasonably OK and fend for myself immediately after, they would probably provide some treatment and take me to a hospital or receive some aid. That is not happening in the dolphin scenario.
Also, many marine mammals who strand are sick. In some situations, dragging them back to the sea is prolonging their suffering because they will get stranded again.
Moments after my reading this thread, I saw ajust-posted news story of how a number of beach-goers in Argentina managed to kill at least one baby dolphin (of species listed as ‘vulnerable to extinction’) by passing it around onshore to take selfies with it. No mention of ‘stranding’ but it emphasizes just how easily it is to damage (fatally) marine mammals out of their element.
IIRC, most of the documentary/videos I have seen of dolphin/small-whales show the use of a hammock-like thing (forget what its called - two long poles with a ‘hammock’ betweenst them) to avoid dragging the animal whatsoever. Again, IIRC, at least one of the marine mammal ‘specialists’ mentioned that dragging them could rub belly raw enough to invite opportunistc infection(s) at areas of damaged tissue. Better to wait for properly supervised carrying and keep animal wet for hydration of skin to help body-temp/skin-drying issues.
I can’t imagine dragging something with very-powerful tail by that tail itself. A human would likely be pummeled if animal was energetic enough to resist.