Tear staining is most obvious on dogs with white and other light color coats or with heavy wrinkles. The stain is usually reddish-brown. 

 Whenever hair rests around the eyes some amount of tear staining results from the hair wicking moisture from the eyes, both above and mainly below the eye. But there are many other sources of tear stains. Tear staining can be traced to health and diet, as well as genetics. Most veterinarians agree that face staining results from excessive tearing. In this case, the damp face hair is a breeding ground for bacterial and yeast growth. The most common is "Red Yeast" which is usually associated with reddish-brown facial stains, and which may emit a moderate to noticeably strong odor. Tear ducts may become infected and result in excess tearing and noticeable staining.

 Some owners have consulted veterinary eye specialists on the problem. The doctors advised them that the eye structure was the most probable source of the problem. If that is so, then genetics would likely play a role and explain why the problem is more pronounced in some pets of the same breed. If you are purchasing a puppy and you care concerned about the potential for tearing and staining, you should observe the mother and sire, and others in the direct lineage.

 Eye duct surgical procedures to increase their tear capacity may help some pets; ask your veterinarian.

 Haircoat stains also occur in areas other than the eyes. White and light color coats, wrinkled faces and short nosed breeds like our beloved bulldog can acquire water stains from pet drinking water, bacteria and red or pimple looking bumps may appear as well, try using a stainless steel bowl for food and water and clean at least 3 times weekly in a dishwasher to sterilize them. Minerals in the water may stain facial hair in the whisker, and mouth areas, as well as other areas on the chest and front legs when water regular drips. Purified waters/bottled water with low mineral content may be the answer.

 Some pet foods with certain additives will stain hair in the mouth area, HOWEVER BEET PULP WILL NOT CASE TEAR STAINS, that is mis-information to assume it will. Beet pulp used in dog foods and treats are "sugar" beets, they are white, not red and do not contribute to tear stains.

 Your first step is to determine the source of tear and face stains. As we have noted, it may be water and food sources and that can be corrected. Tear staining often involves more complex solutions requiring veterinary introspection to determine the source.

 If bacterial and yeast infections are involved you need to take steps to reduce and eliminate their presence. Veterinarians can prescribe medication to treat bacterial and yeast infections. Your veterinarian or eye specialist veterinarians can determine if excessive tearing is the source of stains, and describe alternatives available.

Other Methods: (we do not use these)

Tetracycline 
 Consult your vet for this method as it requires a RX anyway. This method should only be used when all others have failed and should only be used for 1-2 courses of treatment and not relied on as a continued treatment. Some have found success in eliminating tear staining by putting a bulldog on a ten day course of low dose tetracycline. Occasionally this may need to be repeated. Do not use this in puppies that have not yet cut their adult teeth. Tetracycline has been shown to cause teeth which have not erupted to permanently stain yellow. 

TUMS 
 If you need to change the pH of your dogs system to aid in preventing yeast or infection then Tums is primarily a source of Calcium, known as an antacid formulated as 500 mg Calcium Carbonate. Giving your Bulldog 1/2 of the Fruit flavored variety twice a day will help change the pH of the tears. This will change the tears' environment and can help make it hostile for the continued growth of yeast and bacteria.

White Vinegar
 A teaspoon of white cider vinegar can be added to your dogs drinking water to control new tear stains.  It may take a while for your dog to "decide" to drink this water so start with a little less and gradually increase the amount of vinegar.  Vinegar works much like TUMS in that it can changes the pH of the drinking water. Changing the pH of your dog slightly will do wonders in the tear stain war and help eliminate bacteria and deep stain color and prevent yeast build up and is best used as a prevention once you have the tear stains removed or nearly removed. This will help for the future of the tear stains and make them a minimum occurrence.

Milk of Magnesia, corn starch and peroxide  Use equal volume of MOM (plain white) and peroxide, and then use the corn starch to make a good paste of this; put on and work well into the stained area and let dry 4 hours. Wash out, CONDITION WELL. Keep doing this for several days until tear staining is gone, although I would recommend skipping a day or two between applications if possible. Apply a thin coat of DESITIN diaper rash ointment after the area is washed out and dry. Try this every other day if possible and the choice way of doing it. If your dog has heavy stains then do this for 3 days in a row, then skip every other day.
 This MOM formula is my personal preference when I do need to remove tear stain color from the face of dogs. It works well but give it a few days to show results. (Have tried this one, but pain in the butt, I didn't use long enough to know whether it worked or not)

Corn Flour, water and a drop of bleach Take a 2 teaspoon of corn flour, add a few drops of boiling hot water and mix to a paste. Add 1 single drop of plain bleach. mix again. Allow the mixture to cool down and apply to the tear stain area. Allow to dry (10 minutes), wait 1 hour and remove with warm water and clean cloth. Then wipe on a thin layer or peroxide and allow to air dry and leave alone. Repeat every other day for 2 weeks. DO NOT get any in the dogs eye.

 You should also make sure your dog don't have a tear duct obstruction (blockage) and have corrected the food and water issue first so he don't keep getting tear stains. After all if you treat the tear stains to remove them while at the same time you feed a food that causes tear stains you won't gain nothing. Try the least invasive of the above methods first, give them 2-4 weeks to work, then if your not satisfied go to another method of treatment. Do not use multiple treatments at the same time. Always have your vet give the dog a complete exam to rule out any serious eye condition first before trying anything else. The commercial products just don't work very well so it is best to just forget about them unless you have a very mild case of stain.

 Adapted from: http://www.bhejei.com/tearsta.htm