Why does my nested loop throw an exception in Python?

you're accessing the indices from 1 to 101 but the first index of a list is 0, you should iterate with for i in range(100)

def door_traversal():
    arr = [0 for i in range(100)]

    for i in range(100):
        for j in range(100):
            arr[j] = not arr[j]

    count = 0
    for i in range(100):
        if arr[i] == 1:
            count += 1
    return count

if you want to use custom indices maybe you could use a dictionary

def door_traversal():
    arr = {}
    for i in range(1, 101, 1):
        arr[i] = 0
    for i in range(1, 101, 1):
        for j in range(i, 101, i):
            arr[j] = not arr[j]

    count = 0
    for i in range(1, 101, 1):
        if arr[i] == 1:
            count += 1
    return count

You have some pretty fundamental errors in your understanding.

First, lists in python always start with an index of 0. A list of size 3 looks like this:

three = [0, 1, 2]

The indexes are the values themselves. And yes, you could ignore the first element, and create a list like so:

names = [None, 'Wayne', 'CodeMonkey', 'King Arthur']

But it has 4 elements. Everywhere in your code you're going to have to adjust this. Want to know the length? len(names) - 1. And so on and so forth. And if you're splitting your list, you're going to have to add values to those lists if you want consistent behavior:

these_names = names[:2]
those_names = [None] + names[2:]

That's pretty painful. Don't do that - just get used the the fact that the index of an element in the list means the start of the list + index elements. So the item at the start of the list is the list name, e.g. arr, plus [0] elements.

Here's your code re-written in a way that works, under the assumption that everything else is correct.

def door_traversal():                                             
    # If you really want a list with 101 elements,
    # So you can access the last item with `arr[100]` rather than `arr[99]`.               
    # If you're using Python2, `range` is a list                  
    # already.                                                     
    arr = list(0 for _ in range(101))
    # Or use a list comprehension
    arr = [0 for _ in range(101)]

    # There are probably better ways to do this, but it works now 
    # also, no need to provide the step size if it's 1            
    for i in range(1, 101):                                       
        for j in range(i, 101, i):                                
            arr[j] = not arr[j]                                   

    count = 0                                                     
    for i in range(1, 101, 1):                                    
        if arr[i] == 1:                                           
            count += 1                                            
    return count                                                  

print(door_traversal())                                           

Also, I probably would rename arr to doors, or visited_doors, if that's what they represent.

In this line you add 100 items to arr

    for i in range(1, 101, 1):
        arr.append(0)

So your arr array is 100 items long. Then in this loop:

    for i in range(1, 101, 1):
        for j in range(i, 101, i):
            arr[j] = not arr[j]

You say for i in between 1 and 101 you want to run the inner loop over j. The problem is that indices start at zero in Python, so arr[101] does not exist, only arr[100] If you want to ignore the index zero element, that's fine, you can start iterating through the array at index one, but the final element in the array will still be at place 100, as you have only added one hundred items to the list!

You can solve this with:

def door_traversal():
    arr = []
    for i in range(100):
        arr.append(0)

    for i in range(1, 100):
        for j in range(i, 100, i):
            arr[j] = not arr[j]

    count = 0
    for i in range(0, 100):
        if arr[i] == 1:
            count += 1
    return count

Index problems aside, please consider the following python idioms (with comments which are for pedagogical purposes only):

def door_traversal():
    """got doc?"""  # why does this func. exist?

    arr = [0] * 100  # list concatenation

    for i in range(1, 101, 1):
        for j in range(i, 101, i):
            arr[j] = not arr[j]
    return arr.count(1)  # use the list method