Actually, I figured it out. Your algorithm is going the wrong direction.
First of all, not sure why you are looping through the array when you know it will always have only 2 items. You can refer to those as arr[0] and arr[1]. Once you realize that, you only need to create a loop which loops through each letter of arr[1] and see if that letter is somewhere in arr[0]. You are on the right track using indexOf, but you only need to be using it on arr[0] and not on both.
Whenever you are unsure what a specific line of code is doing, console.log() is your friend. I was going to try to explain why your code shows true for everything, but it was complicated and I felt it was better to point a few mistakes out and get you started in the right direction. If you need more help or hints, please feel free to ask.
You will still have a for…loop but you don’t have to loop through an array. Also, I find it easier to work with variables names instead of things like arr[0] or arr[1]. If you want to convert the words to lower case you must assign them to a new variable because strings themselves are immutable.
For example, it might help to do something like this at the beginning of your function:
var lettersToCheck = arr[0].toLowerCase();
var lettersToFind = arr[1].toLowerCase();
Now you have lower case versions of each word. The best part is, you no longer have to refer to arr[0] or arr[1] or remember which one is supposed to be checking which, because your variable names make it clear.