Protecting opamp inputs when powered down
Yes, it will tend to raise the voltage on the positive rail from zero to something else. The situation may be fine for the op-amp if the rail never exceeds a safe value, but it could cause other issues with other devices on the same rail. 

If you want to actually clamp it you should have some series resistance before and preferably after the diode and clamp the rail to some safe voltage if there is insufficient load on the supply to do so. For example, if you can put 10K resistors on the inputs you will only have 1mA flowing with 10V applied so that may be adequate protection without any external diodes at all. 

Keep in mind that an ordinary silicon diode will not prevent significant current from flowing through the protection diodes in the op-amp, the current will be shared to some degree. Adding a series resistor can help reduce the amount that flows through the op-amp inputs. Schottky diodes tend to have less forward drop at the same current so they will hog most of the current, however they are also very leaky in comparison so they can affect your analog circuit.