Yes. The residence can be sold and the proceeds can be reinvested in a new residence. Because a QPRT is a grantor trust, any gain recognized on the sale of a principal residence should qualify for the $250,000/$500,000 exclusion of gain from the sale of a principal residence, provided all of the other applicable tax code requirements are met. The exclusion of gain does not apply to the sale of a personal residence that is not a principal residence, such as a vacation home.
If the proceeds of sale are not reinvested in a personal residence, the QPRT will convert to a Grantor Retained Annuity Trust or "GRAT" and will pay an annuity to the donor for the balance of the QPRT term.