Do universities notify failed candidates? After interviewing for faculty positions, would universities (in the US, or in general) notify me if they chose to proceed with other candidates or to close the search?
Generally, yes, they will let you know, but only after someone else has been chosen and they cannot proceed with your application any further.  But you won't be told that you are low on the list of people, nor likely to be told if you are just a "backup" candidate. 

In the case in which the search is closed with no candidate at all, they may not notify everyone, but I think that varies. 

There are exceptions, of course. And some places will have pretty strict rules and regulations about what is proper here. 

If you think you are in the running and wonder whether you have been rejected, you can ask for an update on the state of the search. I, personally, wouldn't ask specifically about my own chances, but a general update will probably give you information you need to make other decisions. 



There is a caveat here, I guess. If you are in the running for a position, but have other options and let them know that, they may give you more information. But the information is likely to be self serving. Something like: "You are high on our list of candidates, so don't lose heart."