Should the witness falsely testify that the former owner's acknowledgment of vacancy was written before the property was sold to guarantee the tenant's right, or should he tell the truth and the tenant lose his apartment?

It is permissible for the lessor to sell the leased house, and the buyer will own the house with its usufruct waived until the end of the lease term. The usufruct of the lease is excluded from the sale. If what the tenant pays is part of the rent, it will be credited to him, and he is only obligated to pay the remaining rent to the new owner.

As for outright lying in testimony, it is not permissible, even if it is to obtain a right. However, it is permissible to use tawriyah (allusory speech) and ta'wil (interpretation) to obtain a right from someone who denies it. This means that the one taking the oath intends by his words something that contradicts their apparent meaning.