Usually, no. Standard renters insurance generally does not cover bedbug infestations or the cost to replace furniture, mattresses, or other belongings affected by bedbugs. Insurers commonly treat bedbugs and other pest problems as a maintenance or infestation issue rather than a sudden, covered loss.
That said, responsibility for treatment can depend on your lease and state or local landlord-tenant rules. In some situations, a landlord may be responsible for extermination, especially if the problem affects multiple units or was not caused by the tenant. The safest wording is: renters insurance usually does not pay to replace furniture damaged by bedbugs, but you should check your policy and lease for any exceptions or local responsibilities.
The information in this answer is general in nature and is not legal or insurance advice for your specific situation. Policy language varies carrier by carrier; your declarations page is the final word. If you're unsure, call your agent before you need to file.