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This is horrible advice. There are plenty of things you want to delete, like references to elements when you don't need them anymore. A quick search of the Closure Library, jQuery, Backbone, Knockout, Angular.js, and ember.js found many uses of delete. Burn this article in a fire.
As for the supposed performance difference, let's see a jsperf on that.
Edit found one:
http://jsperf.com/delete-vs-undefined-vs-null/6
Delete is a little slower, certainly not enough to warrant the 'never' emphasis.