As with ALL performance questions, they should be examined by actually measuring the performance. In a rather simple test case (your actual code might vary some), I find mixed results across the browsers:

Chrome and Firefox don't differ very much between the two tests, but the slight difference is in opposite directions between the two. IE9 shows the test using a saved copy of this
which I called self
to be significantly slower.
Without a significant and consistent performance difference in Chrome and Firefox and IE9 showing the this
test case to be substantially faster, I think you can conclude that the design pattern you asked about is not delivering a consistent performance boost across browsers.
In my code, I save a copy of this
to another variable only when I need it for a consistent reference to the original object inside of inline event handlers, callbacks or methods where this
has been set to something else. In other words, I only apply this design pattern when needed.
In a previous discussion of this design pattern here on SO, it was concluded that some libraries use this design pattern in order to allow additional minimization because this
cannot be minified below the four characters that it takes up, but assigning it to a locally variable can be minified to a single character variable name.