The site I linked earlier has some information on eye tears and causes. It talks about vinegar and Tums to change the ph. Tums will increase alkalinity, and vinegar increases acidity. I guess it depends on what the dog's problem is. It doesn't suggest apple cider vinegar; it suggests white vinegar.
Tear Staining: The Causes & Cures It states:
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.
Remember, some puppies tear while teething and may outgrow this when they are through teething. Distilled water worked for Joey as a puppy, the kind you use in baby formula, they take out all the minerals, don't use the distilled water that’s meant for irons and appliances, it is not considered safe for drinking.