Turn Off Alt Tags On Links With CSS?

Simple answer: no

They are impossible to turn off just with CSS. They are browser dependant and are not part of any CSS spec i.e. you can't style them, hide them, anything.

Though you can get rid of them by Javascript:

Javascript

var elements = document.getElementsByTagName('a');

for (var i = 0, len = elements.length; i < len; i++)
{
  elements[i].removeAttribute('title');
}

They're actually title tags and you can't remove them with css (you can with javascript) but I guess they're there for a reason.