No, Cymbalta doesn't have any appreciable affinity for your opioid receptors.
However, "Noradrenaline Reuptake Inhibition" generally equals "improved resilience to chronic pain", and Cymbalta gives you a bigger noradrenaline boost than the other popular SNRI, Effexor
Word of caution: Should you take Cymbalta but still require stronger pain relief whenever the pain really flares up, then you should probably avoid Tramadol, or atleast be aware of the interactions: Cymbalta inhibits the conversion of Tramadol (weak opioid, strong serotonergic) to O-Desmethyl-Tramadol (strong opioid, weak serotonergic); in other words: Trams will be weaker on Cymbalta, but you can't just take more of them because you're increasing the risk of serotonin syndrome. Which also means that attempting the recreational use of Tramadol on Cymbalta is basically a trip to seizuresville.