Is moth damage covered by renter's insurance?

No, renters insurance will not cover personal property or loss of use for an infestation of any kind(termites, bed bugs. etc...). You may check with the landlord policy to see if there is any kind of reimbursement through their insurance.

However liability may be covered as shown in this example:

We're going to pretend that your neighbors 10 year old daughter came over for a sleepover with your 10 year old daughter. We're going to also pretend that during the night your neighbor's child was bitten by the bugs so badly that it caused her face and neck to swell. The next day your neighbor is understandably furious, and holds you responsible for their child's medical expenses. You are automatically covered under the "Medical Payments to Others" coverage. If the child misses two months of school, the parents may want to hire a tutor. Your neighbor may also find it necessary to stay home with their child. In that case, they will probably sue you for lost wages and the cost of home schooling their child. This would be covered under the "Personal/Family Liability" coverage.