You could extend the String object with your own function that does what you need (useful if there's ever missing functionality):

String.prototype.replaceArray = function(find, replace) {
  var replaceString = this;
  for (var i = 0; i < find.length; i++) {
    replaceString = replaceString.replace(find[i], replace[i]);
  }
  return replaceString;
};

For global replace you could use regex:

String.prototype.replaceArray = function(find, replace) {
  var replaceString = this;
  var regex; 
  for (var i = 0; i < find.length; i++) {
    regex = new RegExp(find[i], "g");
    replaceString = replaceString.replace(regex, replace[i]);
  }
  return replaceString;
};

To use the function it'd be similar to your PHP example:

var textarea = $(this).val();
var find = ["<", ">", "\n"];
var replace = ["&lt;", "&gt;", "<br/>"];
textarea = textarea.replaceArray(find, replace);

Common Mistake

Nearly all answers on this page use cumulative replacement and thus suffer the same flaw where replacement strings are themselves subject to replacement. Here are a couple examples where this pattern fails (h/t @KurokiKaze @derekdreery):

function replaceCumulative(str, find, replace) {
  for (var i = 0; i < find.length; i++)
    str = str.replace(new RegExp(find[i],"g"), replace[i]);
  return str;
};

// Fails in some cases:
console.log( replaceCumulative( "tar pit", ['tar','pit'], ['capitol','house'] ) );
console.log( replaceCumulative( "you & me", ['you','me'], ['me','you'] ) );

Solution

function replaceBulk( str, findArray, replaceArray ){
  var i, regex = [], map = {}; 
  for( i=0; i<findArray.length; i++ ){ 
    regex.push( findArray[i].replace(/([-[\]{}()*+?.\\^$|#,])/g,'\\$1') );
    map[findArray[i]] = replaceArray[i]; 
  }
  regex = regex.join('|');
  str = str.replace( new RegExp( regex, 'g' ), function(matched){
    return map[matched];
  });
  return str;
}

// Test:
console.log( replaceBulk( "tar pit", ['tar','pit'], ['capitol','house'] ) );
console.log( replaceBulk( "you & me", ['you','me'], ['me','you'] ) );

Note:

This is a more compatible variation of @elchininet's solution, which uses map() and Array.indexOf() and thus won't work in IE8 and older.

@elchininet's implementation holds truer to PHP's str_replace(), because it also allows strings as find/replace parameters, and will use the first find array match if there are duplicates (my version will use the last). I didn't accept strings in this implementation because that case is already handled by JS's built-in String.replace().

text = text.replace(/</g, '&lt;').replace(/>/g, '&gt;').replace(/\n/g, '<br/>');