Stains on the inside of the eyes are a common cosmetic problem in white coat dogs and are caused by the tears. If the tear flow is not excessive and you do not show your dog in competitions then since it is only cosmetic, I would recommend to let it go. It is not worth putting your dog through treatments and the potential risks for this.
Tear stains are caused by an overflow of tears on the face of the dog. Tears are produced constantly to keep the eyes wet. Usually they are collected in the nasolacrimal ducts which collect the exces tears from the eyes and take them into the nose. If there is a problem in the eye region, tears can be produced in excess and overflow on the face. Other problem that can occur is that the lacrimal ducts are blocked or closed and then all the tears wet the face. Statistically 20% of the small dog breeds are born with blocked lacrimalducts. In some cases they can be opened surgically but the pros and cons of this procedure have to be analyised.
Some breed have more prominent eyes which streches the eyelids and cuts the flow through the canals. This is the most common problem and there is little that can be done. Also there is a genetic predisposition toward a high volume of tears. Some may have a reduce lacrimal flow while others may be completely opposite. Diet also plays an important role especially if it is a case of beard stains.
If you have observed an increase in the tears on the face of your dog please take it to your veterinary surgeon to examine it because it can be a serious problem which needs attention.
The cause of the coloured staines is humidity, a condition that allows abnormal bacteria and yeast to develop, particulary Ptyrosporin(Red Yeast) which causes reddish-brown stains. In order to help dry the area the hair under the eyes and on the inside part of them can be cut. If you have not done this before please ask a professional groomer or veterinary nurse to perform this for you. Extreme attention should be given not to damage the skin and more importantly the eyes.
You can wipe the hair under the eyes with different products in order to remove the excess tears. This has to be done at least once a day if you want to obtain some result. The products are a light soap-water solution especially made for dogs. The product should be kept out of the eyes. Please do not use products intended for humans or household products. They are very dangerous for your dog. You can find these products at dog shows, in pet store and at your local vet: Nutri-Vet eye cleaner, Show eyes, etc. Also a fatty acid supplement (omega3) can be administered.