Hi, Lori, and welcome to the FICO Forums!
No, an unpaid CO will not remain forever. By law, it must be removed from your reports no later than 7.5 years from the date of the first delinquency (missed payment) that led to the charge off, whether it is paid or unpaid.
Having said that, many mortgage lenders make paying off outstanding delinquent debts a condition of receiving a loan.
If your mortgage guy is unclear on this, you can refer him to FCRA 605(a)(4) and FCRA 605(c)(1).
