Learn English – “Not… neither… nor…” word order

No, "neither" must always come first in this construction.

Also it generally comes after the verb in the example you give, not before it:

I like neither thing A nor thing B.

As @Richard points out, putting "neither" before the verb implies that the verb is part of what the "neither" is applying to:

I neither like nor detest thing A.

Both of the following seem fine:

Look not for treasure or earthly pleasure.

Look neither for treasure nor earthly pleasure.

The following seems odd:

Look not for treasure nor earthly pleasure.

Consider a plain example:

If I have no pen and no pencil:

I don't have a pen or a pencil.

or

I have neither a pen nor a pencil.

But the negation is perfectly clear in the first example of the two. In fact, that's what we usually say it, right?

I often see sentences such as the following:

I don't have a pen nor a pencil.

I consider this to be maybe a form of hypercorrection. Just a hunch, but people seem to be very apprehensive about the use of "or" recently, perhaps as a result of the growing influence of legalese on regular language.