The discharge and sneezing you describe indicate your dog likely has a condition called rhinitis, which simply means that the mucous membranes within the nasal passages are chronically inflamed. With chronic inflammation often comes periodic secondary bacterial infection, which from the history your dog seems to have experienced on at least one occasion.
There are many potential causes for rhinitis, and it seems as though your vets have tried to rule out a couple of them. In older dogs, dental disease is a common problem, and in toy breeds like the Toy Poodle they are especially prevalent. The teeth cleaning and extractions that Noocher has had would seem to exclude this as the source of the problem. The only other thing to consider in regard to the teeth is the presence of an oronasal fistula. This occurs when a hole forms which allows the mouth to communicate with the nasal cavity. The most common reason for the formation of such a hole is problems with either the canine (fang) or carnassial (large grinding) teeth. Dogs will very commonly have such dental disease bilaterally, so it's possible for fistulas to form bilaterally as well. I would assume that this was checked for while he was under anesthesia for the teeth cleaning. You may want to ask the vet who performed the procedure to be certain.
Another common cause of rhinitis in an elderly dog is some sort of tumor. Generally, however, the discharge would be from one nostril only and since the problem has been such a long-standing one it seems like we can probably safely take this off the list, too.
Your vets have also tried a variety of antihistamines in an attempt to determine if there was an allergic component to the nasal discharge. If Benadryl and Claritin didn't work, you might consider trying a prescription antihistamine called hydroxyzine. Given his pre-existing heart disease, this medication would need to be used cautiously, especially if he begins taking any drugs other than the Lasix. Some dogs can also be particularly sensitive to things in their environment such as heavy perfumes or deodorizers, cigarette smoke, or mold. Carefully examine the home to see if there might be something there that is causing his nasal passages chronic irritation. Eliminating the source of the allergy will eliminate the problem. If someone in the home is a smoker, have them do it outside and away from Noocher.
If all of the above are ruled out than in order to get to the root of the problem, ideally a nasal swab would be taken and cultured to determine if any bacteria, fungi, or other infectious organisms were present. A rhinoscopy would allow a thorough evaluation of the nasal passages and x-rays of the skull might elucidate any bony changes. However, given Noocher's age and the current state of his heart, it may not be worthwhile to put him through all of that. If the nasal discharge and the sneezing are not seriously impairing his quality of life, it may be better just to treat the infections if they arise and always keep a tissue handy.
I will be away from the computer for the next few days, so if you have further questions I can address them when I return.
Hope this was helpful.