No Longer Exists

Solution 1:

Yes, these are both correct, and have the same meaning. However, your second example doesn’t sound very natural; normally, we use do in English to show emphasis, in the affirmative:

— Does the opposition between good and evil exist any longer?

— It does still exist!

If you want to add emphasis to “no longer”, you can phrase it negatively, with “any longer”.

The opposition between good and evil doesn’t exist any longer.

Solution 2:

The opposition between good and evil no longer exists. [normal]

The above is the normal way of saying it.

The opposition between good and evil does no longer exist. [unidiomatic]

Strictly speaking the above is grammatically possible but no-one would ever say it.

The opposition between good and evil exists no longer. [dramatic]

That sounds overdramatic as though you are making a speech.

If you want to use 'to do' then you should say:

The opposition between good and evil does not exist any longer. [emphatic]