My landlord has hired an appraiser to look through the house in which my apartment is one of three units. What happens to my lease if the house is sold?

Nothing. The new buyer has to honor any existing leases. However, they may choose to not renew the lease upon expiration.

You can always ask the current owner to extend your lease if you know you want to stay and your rent rate is reasonable, etc. It should make the property more desirable to prospective buyers to have people locked into longer leases (unless the rent is ridiculously low or something), and you'd be locked into living there and paying the same amount of rent.

Karla in TX