Right-click on New Tab button to open URI from clipboard in new tab. You can use Custom Buttons extension (paste code into "initialization" section) or something like http://userchromejs.mozdev.org/ or https://addons.mozilla.org/addon/uc/ to run this code

newTabButtonRightClick.js

addEventListener("contextmenu", handleClick, true);
addEventListener("click", handleClick, true);
addEventListener("unload", function destroy(e) {
	removeEventListener(e.type, destroy, false);
	removeEventListener("contextmenu", handleClick, true);
	removeEventListener("click", handleClick, true);
}, false);
function isNewTabButton(e) {
	var trg = e.originalTarget || e.target;
	var cl = trg.classList;
	if(!cl)
		return false;
	return cl.contains("tabs-newtab-button")
		|| cl.contains("tabs-newbutton") // SeaMonkey
		|| trg.id == "new-tab-button";
}
function handleClick(e) {
	if(!isNewTabButton(e) || e.ctrlKey || e.shiftKey || e.altKey || e.metaKey)
		return;
	if(e.type == "click" && e.button != 2)
		return;
	e.preventDefault();
	e.stopPropagation();
	if(e.type == "contextmenu")
		return;
	var clipURI = readFromClipboard() || "";
	if(!/^\w+:\S+$/.test(clipURI))
		return;
	gBrowser.selectedTab = gBrowser.addTab(clipURI);
}