jquery preventing hover function on touch

I think a clear approach would be to:

1. Detect if the browser supports touch events
2. Add the hover event handler accordingly

If you're using something like Modernizr already:

if(!Modernizr.touch){
    // only if the browser doesn't support touch events,
    // add the hover handler here.
}
//add the click handler here, as you want it bound no matter what

See What's the best way to detect a 'touch screen' device using JavaScript? and What's the best way to detect a 'touch screen' device using JavaScript? for other options to detect touch capabilities.

Make the .hover() method more explicit and combine it with .on():

var $close1 = $('#close_1'),
    $close2 = $('#close_2');

$('#close').on({
    mouseenter: function(){
        $close2.css({display:'none'});
        $close1.css({display:'block'});
    },
    mouseleave: function(){
        $close1.css({display:'none'});
        $close2.css({display:'block'});
    }
});

Then combine that with .off().

$('#close').on('touchstart',function(){
    $(this).off('mouseenter,mouseleave');
});

If you want the event to fire on click with touch devices, but on hover on desktop devices, then put the functions as a separate function you call within those actions respectively.

EDIT

Been a while since I did this answer, here is a better way:

$(function(){
    var isTouchDevice = ('ontouchstart' in window || 'onmsgesturechange' in window),
        $close = $('#close'),
        $close1 = $('#close_1'),
        $close2 = $('#close_2');

    if(!isTouchDevice){
        $close.on({
            mouseenter: function(){
                $close2.hide();
                $close1.show();
            },
            mouseleave: function(){
                $close1.hide();
                $close2.show();
            }
        });
    }

    $close.on('click',function(){
        $('#full_image').animate({height:0},300,function(){
            $(this).find('img').attr('src','#');
        });

        $close.hide();
        $close1.hide();
        $close2.hide();
    });
});

This doesn't require a "hover prevention" event to fire with each touch, basically sets capabilities on page load while not affecting the click event.