You're probably a
Hence the computer tag.
I am interested in an object that was first created in 1936
You were created in 1936.
I am known across the web
I am unknown to people who try and track me down
An actual, pure Turing machine doesn't exist.
I smile when you look at me
In many pictures you'll see the tape configuration would look like a smile.
I cause protesting, trouble and more thing's
At turing test competitions, there can be conflict because of different interpretations among other problems. For example, https://en.wikipedia.org/wiki/Loebner_Prize#Critisms
For the puzzle title, it makes sense because
The inventor, Alan Turing is the person after which you are named, but yet you don't truly exist.