Do you lose power on long uphills when you ride a full suspension? I ride an old Specialized Rockhopper (hardtail) and I‘ve talked to some die-hard hardtailers who say that you can lose a lot of power on long uphills because of a bit of wasted energy with a full suspension.
Both the first two answers are right on track. Older full suspension bikes often didn't offer a lock-out, or means to adjust/stiffen the suspension. You would bob in the saddle, making it harder to transfer the power forward. You had to fight with it a bit more. Almost all the new rides do offer lockouts, or a variety of adjustments to stiffen or dampen the ride. There's still more flex and movement than a hardtail of course, but I don't think it requires anywhere near the energy it did for the first edition FS bikes. With good technique, and shifting your weight to achieve optimal traction, the FS bikes are well worth the extra weight and minimal energy loss. A long ride down washboard conditions was enough for me to upgrade.
I think that the HT vs FS is not on wether its climbing vs descending, but rather than the type of terrain. If you see a steep climb with loose rocks or gravel, a good full suspension bike will climb a lot better than a hardtail because the rear suspension will provide better traction. If you're going donwhill on a smooth trail, probably a hardtail will work as well as a full suspension bike. Some racers are starting to go full suspension because they don't get as beat with the terrain as a hardtail. Now, entry level full suspension are really not that good, you probably need to enter 1,500 bucks or above to get a good design.