Whoever has the permit normally holds all the cards. If the sign is grandfathered, and the sign is removed, then another cannot go back up. If the sign is legal conforming, then whoever holds the permit can simply move the sign across the fence to the neighbor’s property, by filing a demolition permit and a new application simultaneously, and there’s nothing anyone can do about it (assuming they have a lease with the neighbor). So the person who holds the permit really calls all the shots, unless the landowner is happy with having no rent and no sign.
Most sign companies require a long lease to build the sign, simply because the economics are that they have to make money or they would not build it. On the renewal, it’s really up to negotiation, but a long lease can often be as advantageous to the landowner as the sign company, since you can use a long lease as collateral at a bank,or to improve the property’s value upon sale, and a short lease will not accomplish either.