Although odonates are predators, they themselves are eaten by other creatures. Only a minority dies of "natural cause".

First of all, most larvae don’t reach the adult stage. They get caught and eaten by other (larger) odonate larvae, fishes, frogs and predacious water insects.

When emerging, or just emerged, odonates are very vulnerable and weak. They cannot fly yet and are not able to walk. Some birds, like the Great Reed Warbler (Acrocephalus arundinaceus) seem to know that. They pick the newly born odonates off the reeds they sit on and feed them to their youngsters.

Once dragonflies have survived their emergence, not many birds are able to catch them. Only a few European birds are known to feed on adult dragonflies: the European Bee-eater (Merops apiaster), some swallows and the Hobby (Falco subbuteo) are among them.
Adult damselflies are easier to catch.

Spiders also eat odonates. Damselflies, which are weak flyers, can be found in many spider webs. But it takes a large spider and a strong web to catch a large dragonfly. Large odonates, like the Aeshnids, are strong and not easily caught. Jurzitza (1979) however shows a picture of Anax parthenope, caught in the web of  Argyope bruennichi.
Some spiders don’t make webs at all but do like the Reed Warblers and catch the slow, weak, newly born odonates when resting on the reeds.

Odonates are each other’s enemies as well. Quite often, dragonflies eat damselflies. And some damsels eat other damsels too. Jurzitza (1979) shows a picture of a female Blue-tailed Damselfly (Ischnura elegans) devouring a male of the same species. I once saw a male Anax imperator catching a male Orthetrum cancellatum, which isn’t as big as the hunter, but still big and very strong. But it doesn’t happen very often that odonates catch such big and strong preys. Usually they only dare doing so when they are really hungry: for example when weather has been too bad to hunt for quite a long time (cold, wet, windy).

Other insects that eat odonates are Asilid flies (Robber flies). I once found an Asilus crabroniformis eating a male Lestes sponsa.

But not only animals eat odonates. Carnivorous plants like sundew (Drosera) eat them too. Odonates that want to rest on these plants get stuck on the sticky drops that cover the leafs and die long after. The sticky fluid solves their skin and then their body fluids are transferred into the plant.

Parasites don’t tend to kill their hosts but want to keep it alive so they can stay longer. Many odonates suffer from parasites. Most remarkable are the mite larvae that can be seen on quite many odonates: small red or orange balls that are attached to their wing veins, thorax or abdomen.
Water mites undergo three active stages: larva, nymph and adult. In the larval stage they are parasites. In the other stages, they are predacious. Mite larvae start looking for a host as soon as they are born. When they find one (an almost full-grown odonate larva), they hide under it’s wing rudiment until the odonate larva emerges. Until then, they aren’t real parasites yet but only sit there, wait and travel along with the larva. That’s called forensic parasitism.
When the odonate larva emerges, the mite larva moves to the dragonfly or damselfly and attaches itself to it. On dragonflies, Arrenurus papillator larvae attach to the wing veins, on damselflies to the thorax or abdomen. Other mite species prefer other places.
On some dragonflies, hundreds of larvae were found. Such high infestation rates make the host animal weak, slow and light-weighted. The host dies when there are so many mite larvae on it’s body that the skin bursts!
After growing to many times it’s original size, the larvae wait until the host flies above water. The larva jumps down and changes into an aquatic, predacious nymph.

Other parasites include small worms that live inside the internal organs of some odonates. According to Jurzitza (1979), the varying numbers of these worms are the cause of the enormous movements of millions of Four Spotted Chasers (Libellula quadrimaculata) at a time.

Some humans can also be considered as enemies of odonates: on some Indonesian islands, dragonflies are eaten in soup, with only their wings removed.

Many human activities are threats to odonates: water pollution, river canalisation, etcetera. These problems are well known and well described by others, so I don’t spend too much text on them.

Weather can be a threat, too. When weather is cold, stormy or wet for a long time, odonates can’t hunt and many of them die lacking food. Extreme drought  can cause lakes, rivers, pools and ponds to run dry, killing many larvae.