Do the two partners have the right to dismiss the third partner due to his negligence at work, and what are his financial entitlements resulting from that, bearing in mind that there are no official documents proving the partnership?

Firstly: Partnership is a permissible contract, and Nasser and Saif may separate from your brother whenever they wish, and they are not legally bound by anything.

Secondly: Your brother Muhammad's move with his family from Taif to Mecca or his sacrificing the Ramadan project is none of the partners' concern, because he did it by his own will.

Thirdly: If your brother falls short and the partners wish to separate from him, they may do so. It is better for them to advise him not to fall short.

Fourthly: The partnership contract and its clauses should be reviewed. If there is no contract, then a knowledgeable person whose judgment is accepted by both parties can be resorted to for arbitration.