Tesla Inc.’s plans to save money by closing most of its retail stores and selling all cars online face a big potential hurdle: Store landlords show no signs of giving the company an easy way out of its leases.

The electric car maker has offered few details about which of its 106 stores and galleries in 26 states it plans to close. Even many shopping center landlords say they have heard nothing from the company about whether their Tesla store will be shut down, according to real-estate brokers and landlords.

What...