What happens if the lease has already technically expired by the time my sale closes?

An expired lease you're still operating under generally puts you in a holdover position, where you continue paying rent and the landlord continues accepting it, often on a month-to-month basis or under terms implied from the old lease — but a holdover tenancy is a much weaker, less certain position to hand off to a buyer than an active lease with time left on it. There may be no formal document left to "assign" at all, since the original term has already run out.

This matters enormously to what your buyer is actually purchasing: instead of a secure leasehold interest for a defined remaining term, they may be inheriting an arrangement the landlord could potentially end on comparatively short notice, depending on the nature of the holdover and what, if anything, was agreed after the original term expired.

Before marketing the business or setting a price, clarify the actual status of the tenancy — whether a new lease was ever signed, and what the landlord is prepared to offer the buyer directly, rather than assuming the old lease can simply be handed over. A Treadstone business lawyer can sort out what's actually happening with the tenancy before you go further.