It is possible for a shark to have no teeth, either by not having teeth to begin with or through losing them. There are a few types of sharks which fall in the former category, such as whale sharks and basking sharks. Both of these are filter feeders that strain their food from large volumes of water, similar to baleen whales.

While a shark which normally does have teeth could conceivable lose all of them, this would take an uncommon event. The toothlessness wouldn't last too long either (I think, and assuming the shark can survive for some time without teeth), since sharks constantly lose and replace their teeth throughout their lives.

There are a number of shark species that do not have teeth. For example, the basking shark is the second largest shark in the world and survives on a diet of plankton. It swims through the water with its giant mouth open and filters the plankton from the water with "gill rakers". Isn't it amazing that a shark as large as a school bus survives on creatures that aren't much bigger than a tip of a pencil!? And all without the need for teeth!

Theoretically, yes, but all of the sharks that I am aware of - even things like whale sharks and manta rays - do have teeth.

All sharks (that I know of) have teeth, unless they have some kind of disease or health problem which is causing them to lose teeth or not grow any (and any shark without teeth would probably starve quickly since it would not be able to hunt), but some sharks may have specialized teeth which do not really look like teeth. It depends on what they eat!

Sharks like the sand tiger shark which eat fish have long, skinny, pointy teeth designed for catching their slippery prey. Sharks like the Epaulette shark which feed on invertebrates on the sea floor have plate-like teeth which are designed for crushing the shells of they prey. Sharks like the great white shark, which eat seals and other mammals, have serrated teeth for ripping out chunks of flesh from their prey. And finally sharks which feed on krill and plankton like the basking sharks have tiny teeth which they don’t even use - instead they filter water through their gills to catch prey.