There are some drugs that can do damage to your hearing, but I don't think there's any evidence that Cymbalta causes damage. Not that it necessarily matters -- tinnitus can be a serious quality-of-life problem. Unfortunately, there's no way to really predict how this will play out for you. I was recently looking up anecdotes of drug-induced tinnitus with Wellbutrin, although I ran into a few from other antidepressants (including Cymbalta). I'll share what I could find, FWIW.
People who had this problem usually got it either when they started the drug or increased the dose. There appears to be a dose-dependent effect. In many cases, it went away within a few days of stopping the drug. But usually the people who were posting were the ones that had longer-term problems. Most people who stopped the drug said that the tinnitus lessened or went away with time (ranging from weeks to months -- there was one report of someone who got better after 2 years). Most of these people had discontinued the drug shortly after they developed tinnitus.
Other people found it to be manageable, or manageable at a lower dose, and continued their medication. Some of them had less tinnitus after taking the drug for a long time, and others had no change.
Out of the 35 reports I documented in my spreadsheet, four people had said they'd had their hearing tested. One said their hearing had been affected, and the other three said hearing was normal (but the tinnitus itself was agonizing). There was another person who said their hearing was worse but had not been tested.
The practical thing I was able to find, though, was that there are things that aggravate it -- particularly anxiety, alcohol, and caffeine/stimulants. (Other things reported to make it worse were NSAIDs, Ambien, loud music/noise, raised blood pressure, headache, lack of sleep, and lying down.) People found that avoiding those things when possible was helpful.
Again, I wasn't actively searching for anything other than Wellbutrin, but perhaps something here helps in some way.