Advice- Should this dog be put down?

My aunt has a toy poodle, he's very old(12-14 years) and has this grey-ish mass growing on and all around his teeth. He's had it for years but they say they can't afford to treat it. It's very smelly.Now their family is planning on moving to an apartment and they are looking to put him down(conveniently they think now that they're moving it's his time). They ignore the dog most of the day. I've noticed when I come home with them he doesn't even get out of his bed to greet them. He's old so it's understandable BUT the poor dog loves me and I love him Whenever I'm over there(everyday) He greets me and runs around me and even jumps(or tries to) for me to hold him which I do ..I know that they don't like him. They love him(I think) because they've have had him for around 10 years but they don't pay any attention to him at all. His eyes accumulate black stuff and sometimes when I come over one of his eyes is stuck completely shut and they don't even notice.So I want opinions. Should he be put down? He's old and I don't think he would live more than 1-2 more years but I really have a problem with their reasoning. He would be happy if I took him. I know he would since I take care of him when they go on vacation and he always seem very lively at my house with me and my dog(though my dog thinks he's annoying lol)I really want your advice