Okay... newbie here.

I know that command centers, fortresses, and turrets do NOT need to be powered by a power core.

But what about the processing centers for each resource?

I've played with them both powered and not powered, and didn't notice a significant difference in production... but I'm guessing that's because I suck either way.

I know it wouldn't effect how much the workers can hold, so perhaps it speeds up the drop-off time or something. If it's true that they DO need power, can someone tell me the difference in production rate?

Yes, they need to be powered, and there is an enormous difference in production. Put 10 banthas into an animal husbandry center that's not powered. It gives you 15 food/min. Now power it: 60 food/min.

All buildings other than the CC, the Fortress, and Turrets, need power, or they will develop technologies and build units at 1/4 the powered rate. Unless you feel like waiting for 10 minutes just to get the carbon processing upgrade, you should power all buildings that need it.

However, you only need to power one of each processing plant (other than the husbandry center). Since you can only develop one technology per tech level, the others really don't need power. It does not affect how fast the workers gather... just how long it takes for your technology research to be completed. So power one of each, and leave therest be.

yeah what chesseck said
but are you talking about food prosecrs or the shelater you put the animals in

I don't know the answer, but I do the question. He is talking about processing centers. Yes, you do because they research techs.