My JS is compiled from higher-level source; at the source level there is no duplication and it's simpler to use a variable. Moroever, using getElementById is 60% slower than a global variable in FF and more than 10x slower in IE. For the app I'm building, this matters a lot.
Of course it's faster to use a precomputed reference than to find an element in the DOM. I can't see the "overhead" of manually storing a reference to a DOM element in a variable ever mattering in any application. Grab it once and you're done.
If you want an element by ID, that's what document.getElementByID is for (or $ if you're using a JS framework). If you want a form element by its name (why not it's id?), you should go out to get that specifically.