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Hahaha genius. "...What??" Oh, and I have no idea what the bug is.
You're checking for a null local variable value in the third conditional, but appending to that same variable in the first two, which would either cause a runtime error in most languages or never pass the third conditional.
This was my first thought as well. If he instead initializes:

    var result = "";
and checks for:

    if (result.length === 0) {
    ...
in the third conditional, is there still a bug?
Yes, he specifies the conditional is "examine the variable result; if it is still null ..."
In Ruby, for example, you could append a string to nil by interpolating the nil into a string literal, or various other ways. Seeing as he didn't specify the language, what the variable was initialized to, or what exactly "append" means, we can only assume that these imaginary things behave as intended.
It might have been when he initialized the variable result.

He goes on to append strings to this variable which would lead me to believe it has some value that you can append to.

This would make the third conditional where he checks for null to fail, he should check for an empty string.

This is my conclusion as well. To wit, an approximate translation of his method into Ruby:

  fizzbuzz = (1..100).to_a.map do |i|
    result = "" # 1:23 initialize a variable
    result += "Fizz" if i % 3 == 0
    result += "Buzz" if i % 5 == 0
    result = i.to_s if result.nil?
    result
  end
The corrected solution would be to change `result.nil?` to `result.empty?`, i.e. checking for "empty string" instead of "null".
The bug in the interview is not having a whiteboard or paper or a computer that the interviewee can use.

Forcing anyone to do a coding exercise in their head is ridiculous.

That video is so fucking annoying. Why not just show a page with the fucking text?
Shouldn't he check the divisibility by both 3 AND 5 first?
No, in that case the first two conditionals will both pass and the result will end up as "FizzBuzz".
Well, you can check for 3 AND 5 first (or check for 15 altogether). This is if you have two subsequent checks for 3 and for 5.
Well, that would depend on the print statement. Generally, all of them have new lines. Even if they don't you would have to generate new lines appropriately otherwise the next number would print right next to Fizz.

PS : My assumption is that you need to print each number or Fizz or Buzz or FizzBuzz in a separate line.

I see paper and pen/pencil in that office...

He doesn't ever print out result either

what? for this task, programmers should be able to come up with code optimized (if not optimal) right away. For example, this line "I would append a string [...] to the result" certainly involves at least one memory copy operation, especially since he suggested doing this for mod 3 and mod 5. Everyone should know that on most CPUs, arithmetic operations are at least an order of magnitude faster than memory operations. So the alternative should be obvious: don't copy the strings into a string buffer. If you just print the 2 strings ('Fizz' and 'Buzz') as necessary (inside the conditional test blocks) the compiler/interpreter will allocate space for these 2 strings ONLY and will pass pointers to them.

Even though the last operation he mentioned "I would check if the string is null" would indeed be fast, the program won't be able to take advantage of it, because by that time, the program will be performing very costly memory operations.

Instead, this involves no extra memory copy:

for i in 1 to 100: if (i mod 3): print 'Fizz'; if (i mod 5): print 'Buzz';

    if (not (i mod 3) and not (i mod 5)): print i;

    print "\n";
Even though the 3rd condition involves a few extra arithmetic/logical operations, these are way faster than memory operations. The strings 'Fizz', 'Buzz' and "\n" are allocated at the beginning of the program. There is no "result" variable that needs to be allocated/deallocated on every iteration and there is also no need to store the result in an array, because the task is to write a program that "prints" the numbers from 1 to 100. Plus, wouldn't you want to store it in a string, instead of allocating memory for arrays?

I would expect good candidates to be able to at least arrive at this solution after their first attempt.