Acknowledge previous advisor/collaborators for ideas in article A PhD student switches to another PhD program and writes an article based on an idea or proposal on which the PhD student has worked, but which was written by the former advisor. 

The PhD student leaves in good terms, and then contacts 2-3 times the former advisor to know whether they are interested in contributing to the article, without never getting a response. 

If the PhD student publishes said article, should a note be added to the acknowledgments mentioning something like "we would like to thank Dr. Smith for some of the ideas that generated this article"? 

Note that none of the original proposal is part of this article but some similar ideas are mentioned.
Acknowledging the contribution of someone, however small it is, is seldom a bad idea.

Usually, when I acknowledge the contribution of someone, I send a copy of the paper to acknowledged people before the final submission, informing them of the acknowledgment. In possible critical cases, I ask whether they are ok or not with the formulation of the acknowledgement, specifying by which date I intend to go on with the submission.

I think that the above approach can be used also in your situation, and should there be no reply by the specified date, I would keep the acknowledgment as it is.