First of all, your code has a little flaw. It will print the first two lines like this:

    ◼◻◼◻◼◻◼◻
    ◼◻◼◻◼◻◼

You may haven't noticed it, because it's hard to spot using a whitespace. This happens, because you use an `else if` when appending the newline:

    if (i !== 0 && j === 0) {} else if {}

 Simply separate the two statements and it will work.

Personally I would move this part to the end of the loop, as it reads like:

> If the line is complete append a line-break.

Also, this test should be sufficient and is maybe easier to read:

    if (size - 1 == j) {
        result += "\n";
    }

<hr>

Then you want to print a field of the chessboard in any case. So the second test is actually redundant:

    if ((j % 2 === 0 && i % 2 === 0) || (j % 2 === 1 && i % 2 === 1)) {
        result += " ";
    } else if((j % 2 === 0 && i % 2 === 1) || (j % 2 === 1 && i % 2 === 0)){
        result += "#"; 
    }

… can become:

    if ((j % 2 === 0 && i % 2 === 0) || (j % 2 === 1 && i % 2 === 1)) {
        result += " ";
    } else {
        result += "#"; 
    }

<hr>

Now, the condition is still hard to read. Let's simplify it to this:

    if (0 === (i + j) % 2) {
   	   	result += "◼";
   	} else {
   	   	result += "◻";
   	}

This is now so simple, that you could easily use the *ternary operator* and rely on JavaScript casting the result to boolean:

    result += (i + j) % 2 ? "◻" : "◼";

<hr>

This will be the result:

<!-- begin snippet: js hide: false console: true babel: false -->

<!-- language: lang-js -->

    var size = 8,
        result = '';

    for (var i = 0; i < size; i++) {
        for (var j = 0; j < size; j++) {
            result += (i + j) % 2 ? "◼" : "◻";

            if (size - 1 == j) {
                result += "\n";
            }
        }
    }

    console.log(result);

<!-- end snippet -->