How to remove Firefox's dotted outline on BUTTONS as well as links?

button::-moz-focus-inner {
  border: 0;
}

No need to define a selector.

:focus {outline:none;}
::-moz-focus-inner {border:0;}

However, this violates accessibility best practices from the W3C. The outline is there to help those navigating with keyboards.

https://www.w3.org/TR/WCAG20-TECHS/F78.html#F78-examples

If you prefer to use CSS to get rid of the dotted outline:

/*for FireFox*/
    input[type="submit"]::-moz-focus-inner, input[type="button"]::-moz-focus-inner
    {   
        border : 0;
    } 
/*for IE8 and below */
    input[type="submit"]:focus, input[type="button"]:focus
    {     
        outline : none; 
    }