When comparing dextromethorphan to the opioid antitussive (cough suppressant), codeine, it is often said that, while codeine is the more potent antitussive, the advantage of dextromethorphan is that it has no opioid-associated abuse potential. This is true but, unfortunately, detromethorphan is not completely free from potential for abuse. 

Codeine is a weak opioid agonist. Low doses of codeine are sufficient to achieve the antitussive effect. Therefore, there is relatively little risk of abuse of codeine when used as an antitussive.  However, up 15% of codeine is metabolised to morphine,  which is a much more potent opioid agonist (1).  Codeine has a well-known potential for abuse and abuse of codeine cough mixtures is a peristent problem.

Dextromethorphan is not an opioid receptor agonist and so does not have opioid-associated abuse potential. However, dextromethorphan is abused as a recreational drug. At very high doses, well above the label-specified maximum dosages for use as an antitussive, dextromethorphan acts as a dissociative anaesthetic.  Like other dissociative anaesthetics, such as ketamine and phencyclidine (PCP), dextromethorphan is abused.  The mechanisms of the dissociative anaesthetic effects of dextromethorphan are thought to involve actions as a nonselective serotonin reuptake inhibitor and an NMDA receptor antagonist.

References:

(1) “Codeine and Morphine Pathway, Pharmacokinetic”
https://www.pharmgkb.org/pathway/PA146123006 [accessed 2nd Feb 2017]