What is the legal ruling in Islam regarding someone who bears the rent of a commercial shop and the subscription for an unused telephone line, while the partner refuses to sell the shop or reopen it, and the costs continue to be paid without any benefit?

The questioner has the right to dissolve the partnership and sell her share, as partnership is one of the permissible contracts. If the other partner refuses to dissolve it, he bears the rent of the premises and the phone charges after the request for dissolution, unless the lease term is still valid, in which case they share in that. If the partnership is dissolved, the agency is terminated.