I respectfully disagree. "I neither want nor need beer" . Here, 'neither' is associated with the 'want' not the object of the want i.e. 'beer'.
"I do not want either beer or wine"?
To my ear, "I want neither beer, nor wine" sounds very literary or archaic - it would be ok in: "I want neither beer, nor wine. Fetch me a flagon of the finest armagnac in Albion or I shall have you flayed!"