Is there any assumed guarantee on dental crowns? A few years ago, my former dentist put a crown on one of my molars. Recently, a part of the crown (near the gum line, between teeth) broke off, and food is getting trapped in the resulting gap, causing gum

There. There are no guarantees expressed or implied with any medical procedure, and it is unethical for any health care provider to do so. You do not say how old the crown is, but it is possible that something is breaking down under the crown. It could also be that nothing broke off, but in fact, you have a bone loss in that area resulting in a space and a bacterial/food trap. Was a recent radiograph taken? What did your current dentist tell you? This cannot be left untreated. From your description, and assuming you are correct that something did break off, it could be that a small piece of porcelain chipped off the crown. If the gold coping under the porcelain flexed due to a cavity in that spot, the porcelain could flake off. If there is a cavity, it needs to be taken care of. Even if no cavity, the food and bacteria being trapped will constantly keep the gum tissue in that area inflammed, which can lead to a periodontal defect. You should see your new dentist to determine what exactly the problem is and what would be the proper course of action.
Crown longevity. I recall a national speaker was advocating guaranteeing dental work. But as the panel has elegantly mentioned, since the crown (or any dental work) is subject to all kinds of oral environments, different bacteria, functions, eating habits, genetics, etc. It's impossible to set parameters for guarantee for any dental work. Your best action is to speak to your dentist and find a happy solution.
No. While there is no guarentee, there are some clinicians that provide for a limited warrenty, usually by keeping up with regular exams and xrays. An abupt failure shortly after placement I will usualy replace, 4 years with no exams or xrays is a different story.
No. As others have mentioned, medical/dental procedures are never guaranteed. However, if you are a regular patient going in for your regular check ups, most drs will be willing to work with you on a solution.
No. There is no guarantee on medical procedures. This is because it is be used an tested in ways the dentist cannot control, and it is in a biological environment. Please speak to your dentist about your concerns and take care...
Tough to say. After a few years of the crown performing well in your mouth, it's hard to say that there was something wrong with it.
No. It is illegal to warranty medical treatment. I used to guarantee my work in writing. Until a dentist reported me to the state board.