Is there any non-pathogenic gram negative bacteria?
As Canadianer already mentioned in the comments, the majority of bacteria (this includes gram-negative bacteria) are non-pathogenic.

I think a prime example is the human gut flora. You can find loads of bacterial species living in your gut without doing you any harm, if anything they are doing you a favour by breaking down food. 

Here some examples of gram-negative genera:


Bacteroides
Escherichia


Most species within those genera are non-pathogenic, however, there are exceptions.