Luckily - the mule - although genetically disposed to possibly inherit most of the equine diseases and problems of the horse, has hybrid vigor which prevents or lessens some of the more common ailments.  There are still diseases which may affect the mule, and also steps of common sense and preventive measures to avoid them.  

EIA - Equine Infectious Anemia - This disease will affect all equine regardless of species, from donkeys to horses to mules to Zebras.  It is highly contagious and spread by the exchange of fluids - usually blood by a biting fly from an infected animal to other animals.  The disease is not always fatal, but the infected animal will carry the disease and be capable of infecting other animals for the rest of its life.  The quality of life may also be decreased, as the animal will be anemic and suffer from the anemia. EIA infected animals may be kept, by quarantining them for the remainder of their lives.  They must be branded by the authorities.  They must live in fly-proof quarters, more than 200 yards away from any other equine.  A sad and lonely life for any herd animal.  
            Infected animals can be determined by regular Coggins tests.  A negative Coggins means a disease-free animal.  Never buy an animal without a recent , negative Coggins test.    Even the Coggins is by no means a 100% proof screening.  They are usually only required once a year.  However, if your animals are doing a great deal of traveling, especially during fly season, you might find it more prudent to test more often.
          How important is it to test for and destroy EIA positive animals?  Case in point, the US BLM had a herd of Mustangs in which one animal was tested positive.  12 months later the herd was brought in for testing, and not 67 of the 100+ animals were positive.  Destruction of the entire herd is pending.  There is unfortunately no cure and no prevention for this disease - just careful management.  Fly care, control of the fly population, and routine testing are the only way to help keep this occurrence low.   Colic - In general - a bellyache. 
         Horses, donkeys, mules - all equine - cannot belch.  They can certainly expel gas the other way, but if there is intestinal blockage, that gas build into a gut-wrenching pain.  It usually occurs from changes in feeding - overeating, major feed type variation, but there can be other factors.  The equine tries to roll to alleviate the pain.  In many cases, the gut (already swollen with gas) begins to displace and possibly slip between other organs.  The intestine may become twisted (torsion) and if care and probably surgery is not done immediately, the twisted portions of gut will die.  Calling the vet at the very first signs of colic may not ensure that the patient lives, but the changes are better than they might!  Contrary to popular belief, mules and donkeys CAN colic, although it is reported far less frequently than in horses.  

FOUNDER - Laminitis.  Certainly donkeys can founder, and it is usually linked to overfeeding. There are any number of factors that can cause founder - too-rich food (such as a bowl of 27% protein dog food!  or a pasture full of fallen acorns), cold water given to an overheated animal, fever after any infection.  The sensitive laminae of the hoof become swollen and hot, causing enormous pain.  Imagine your smashed thumbnail, but wrap that thumbnail all the way around the end of your thumb.  Treatment with drugs and cool washes can help.  Your vet and farrier are your best consultations.  Unfortunately, chronic founder is common - recurring after hard work, spring grass, etc. 
        Abbesses (which can rupture through the coronary band, the sole, or even through the hoof wall) are often seen with chronic founder.  Once foundered, the animal will need special hoof care and restricted diet to avoid recurrence again and again every year.  In severe cases, the hoof walls will continue to grow deformed, and the coffin bone (inside the hoof) may drop or rotate, causing pain and lameness for the rest of the animal's life.  Treatment is only symptomatic - prevention is the only cure. 

         Sarcoids - Donkeys and mules seem more predisposed to Sarcoid tumors than do horses.  Sarcoids are unsightly, but no cancerous tumors.  They often occur on the shoulder, hip, or eyelid.  They must be surgically removed, and cryonically freezing them may prevent their recurrence.  Some will regrow anyway, needed further treatment.  Sarcoids should be removed, as they will not only continue to enlarge in size, but they are easily damaged and will bleed (not to mention being horribly unsightly!!!)  There is no known prevention.  
        Fatty Tumors - Donkeys carry fat more prominently over their body than horses.  Mules also get fatter and carry cellulite deposits, especially over the neck and pones over the hips.  But occasionally tumors occur in gelded donkeys and mules that cause them to look like intact males again.  Why this happens is yet unknown.  They must be surgically removed. Other tumors may also occur in mules, some cancerous, some not.  No known cause can be pinpointed, and therefore there is no prevention and no cure if surgery fails to rectify the problem.  
       Tetanus - Lockjaw, as it was called in the turn of the century, is harbored in placed devoid of oxygen.  It occurs mostly in deep puncture wounds, where it can breed in the depths of the tissue.  Humans and equine alike can contract tetanus - and it is not a pretty way to die. Locking of the muscles, stretched neck, tremors, sensitivity to light and loud noises are all symptoms.  It may take from seven days to several weeks to manifest.  Fast administration of a tetatanus toxoid can prevent further advancement of the disease, but left untreated, it is fatal.  Tetanus can be prevented by annual boosters.   Flu, Rhino, Viral infections - there are vaccines available for equine to prevent or al least offer increased immunity to many virus' and diseases including Rhino, which is an upper respiratory disease. 
        Although the vaccines are labeled for Horse use, there are no variants developed for mules or donkeys.  It is the general consensus of the veterinary practice that any protection offered by these vaccines is better than none.    Rabies, strangles, EPM (possum fever) Vesicular Stomatitis - all of these diseases make the headlines alarmingly each season.  Why - probably because simple preventive measures were not taken.  Many vaccines are available, dewormer can help with fly control by halting the larval cycle of some species.  All it takes is a yearly booster for most animals.  Broodmares should be kept on a schedule recommended by your vet, as should young stock.  Care and prevention are simple and far less expensive than the vet bills you will have for an animal that contracts a life-threatening disease. 
      The other most important factor in care of any equine is quarantine new arrivals - no matter how healthy you feel they are.  You may not be able to do anything for the animal who develops cancer - nature has taken over in that case, but you can certainly try your best to make sure your long (or short-) eared friend has all his shots against disease, wormed, trimmed, and his pasture kept clean!
        There are many excellent vet manuals and books available about the diseases and medical problems in horses.  Almost all of these will cross over to donkeys and mules.  These, with information from and consultation with your vet, are your best sources of knowledge for keeping your mule's quality the best he deserves.