Is it permissible for a sister to sell her share of the inheritance to her brothers, and is it considered unlawful for the brothers to ask her to sell her share if she procrastinates in selling and prefers to receive the rent?

It is permissible for an heir to sell their share, provided there is mutual consent, based on the Almighty's saying: ﴿unless it be a trade by mutual consent among you﴾, and the Hadith: "Indeed, selling is by mutual consent." So, if the sister agrees to sell her share, there is no objection. However, if she procrastinates, she cannot be forced, nor should she be pressured. Patience and support should be extended to her, as her continued benefit from the rent is more advisable than selling her share, especially with her extravagant husband who might squander the price of her share. In summary: The sale is only valid with mutual consent, so do not pressure your sister, and consider what is in her best interest.