Contrary to audiophile wisdom, crossover networks "soak up" almost no power, even when all the drivers are connected. If they did they would get hot, and if mounted in a sealed enclosure stuffed with fiberglass or wool, very hot.
Yes, inductors and capacitors do not dissipate significant power. However I would think that there would be some sonic benefit with many amplifiers to disconnecting the woofer's crossover elements, because the reactive (inductive and capacitive) load seen by the amplifier would be minimized.
Disconnecting the load on the low pass leg may affect characteristics of the high pass leg.
I believe that is only true for the relatively small number of speakers which are designed with the crossover legs in series. In those cases, a low pass filter is connected across the high frequency driver, a high pass filter is connected across the low frequency driver, and the two filters (and consequently the two drivers) are connected in series. The unwanted frequencies for each driver are shunted (bypassed) around the driver. But that kind of design is uncommon, I believe. More typically, the multiple crossover legs are in parallel and do not interact.
Koestner -- If you were thinking of leaving the low pass crossover elements unused, I'm not sure how you are planning to make the rolloff of the signal applied to the woofer complementary to the rolloff of the high pass part of the crossover. I think you would have to use an electronic crossover ahead of the low frequency amplifier, set to provide a low pass function for the woofer with rolloff carefully matched to what those crossover elements originally provided.
Regards,
-- Al
