Python Text-To-Speech program
DRY

Apply the DRY (Do not Repeat Yourself) method. For instance, the check to remove the file. Since it's being called in two places, we can make this a function. I've re-written your code in a more functional style. This is also easier to read what's occurring.

else

Your else: None statements are not doing anything.

continue

Same goes for this in the while loop. This happens anyway.

if

When you check the user's answer, just assume it's going to be varying. Instead of checking for an UPPERCASE 'y' and lowercase, just lower it. Also stripping it to make sure to remove erroneous spaces.

answer = input("Do you want to repeat? (Y/N) ").strip().lower()
if answer in ['yes', 'y']:
# instead of
answer = input("Do you want to repeat? (Y/N): ")
if answer == "y" or answer == "Y":


Exceptions

Since you're interacting with the user from the interpreter, you can catch if the user exits the program and run any cleanup code before cleanly exiting. I've wrapped the loop of the interaction in a try/except statement so that this can be caught.

if name == 'main':

Check this SO post out. 

Rewritten

import webbrowser
import os
import time
import getpass
from gtts import gTTS
from mutagen.mp3 import MP3

my_file = "C:/Users/%USERNAME%/Desktop/TTS/bob.mp3" #Sets a variable for the file path.

username = getpass.getuser() #Gets the username of the current user.

def remove_file():
    """Checks if myfile exists and if so, deletes."""
    if os.path.isfile(my_file):
        os.remove(my_file)


def play_tts():
    webbrowser.open(my_file)  # Opens the .mp3 file
    audio = MP3(my_file)  # Sets a variable so that the Mutagen module knows what file it's working with.
    audio_length = audio.info.length  # Sets a variable of the length of the .mp3 file.
    time.sleep(audio_length + 0.25)  # Waits until the file has finished playing.
    os.system('TASKKILL /F /IM wmplayer.exe')  # Closes Windows Media Player.
    time.sleep(0.5)  # Waits until Windows Media Player has closed.


def ask_and_play():
    # Takes the user's input and uses it for the Text-To-Speech
    tts = gTTS(text=input(
        "Hello there " + username + ". This program isn\n"
        "used to output the user's input as speech.\n"
        "Please input something for the program to say: ")
    )
    tts.save('bob.mp3')  # Saves a .mp3 file of the user's input as speech.
    play_tts()


def check_continue():
    """Checks if the user wants to continue.
    Returns a boolean value."""
    while True:
        answer = input("Do you want to repeat? (Y/N) ").strip().lower()
        if answer in ['yes', 'y']:
            return True
        elif answer in ['no', 'n']:
            return False
        else:
            print("Sorry, I didn't understand that. Please try again with either Y or N.")


def repeat():
    """Repeatedly ask the user to type text to play,
    and check if the user wants to repeat or exit."""
    while True:
        remove_file()
        ask_and_play()
        if not check_continue():
            raise KeyboardInterrupt

if __name__ == '__main__':
    try:
        repeat() #Calls the function.
    except KeyboardInterrupt:
        # clean up
        remove_file()
        print('Goodbye!')