As saikiran.vsk suggested in FF the you can use
select {
-moz-appearance: none;
text-indent: 0.01px;
text-overflow: "";
}
and in IE this will work , but IE 9+
select::-ms-expand{ display:none; }
you can add this either globally or class based.Thanks How to hide drop down arrow in IE8 & IE9? @spudley