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ADA4177: If the inputs range beyond the supply voltages, do I need to concern myself with the load current (that is, the total power of the device)?

Yes. When either input exceeds the rail by more than a diode, large currents flow through the protection JFETs which causes the channels to pinch off and effectively raises their resistance. The ADA4177 will dissipate heat according to the thermal resistance (θJA) of the package it is in, which, in turn, heats up the die. The die temperature must not the absolute maximum rating of 150°C for device protection. Extended over-temperature exposure can cause some operating specifications to shift outside of their guaranteed limits.