You need a space between [[ and " and also between "and ]]. You can also drop " around your strings when using [[ and ]] to test - but leave the space there. Otherwise, bash will do this:
[[yes == Yes
With the response:
[[yes: command not found...
The || makes it try the next part and it's here that it executes the command that becomes the "magical" loop:
yes == yes
yes is actually a command that will repeat what you give it as an argument (it defaults to y to be able to do yes | program to answer all questions that program asks with y). It will continue to repeat the same thing until stdout is closed (the receiver closes stdin).
So, it will print == yes in an endless stream.
Fix:
#!/bin/bash
echo "Found a previously saved file. Do you want to continue with this? (Yes/No)"
read -r Response
if [[ $Response == Yes || $Response == yes || $Response == y || $Response == Y ]]; then
    echo "Success" 
else
    echo "Fail"
fi