You should declare sum and result outside for loop. Try this:
function fibonacciSum(){
var i;
var fib = new Array ();
fib[0] = 0;
fib[1] = 1;
var sum = 0;
var result = 0;
for(i=2; i<=10; i++){
fib[i] = fib[i-2] + fib[i-1];
var number = fib[i];
if (number % 2 == 0) {
result = sum += fib[i];
}
}
console.log(result);
}