"I love everyone very much."
Translation:Herkesi çok seviyorum!
"Herkesi" is in the accusative case, and is used when "everyone" is the direct object (as in this sentence).
"Herkese" is in the dative case, and is used when "everyone" is the indirect object: when something is being said to everyone, given to everyone, done for everyone, etc. ("I'm buying coffee for everyone." -> "Ben herkese kahve alıyorum.")
It's receiving the action as a direct object, though. Herkes is the thing you love, so it's the direct recipient of your action, and takes the accusative case.
The dative case would be for an indirect object: not the direct recipient of the action, but the person (or thing) for whom you perform the action.
For instance, if I buy coffee for everyone, buying is the action. The coffee is the direct object: it's the thing that I'm buying. Everyone is the indirect object: I'm not buying everyone (I'm buying coffee), but I'm buying FOR everyone. So in that case, "everyone" would get the dative case.
(I hope that makes sense. It's confusing.)