Seeing as pretty much all rage-based champions are underwhelming on Dominion, can take a look at the Dominion aura and possibly adding rage decay reduction (and possibly increasing rage gain) or removing it altogether for the mode.

As for energy-based champions it'd be a bit harder, as an energy regen aura on dominion would be a direct buff to all the energy champions, including the already viable ones.
I guess adjusting each and every energy based champion or ability individually is not Riot's style, so that's out the window too.
Modifying health relics to give energy on top of mana/hp could be a step in the right direction here.

Removing or nerfing the global mana regen would also indirectly buff rage- and energy (and cooldown only-)based champs.

All of the champs that struggle with rage or energy issues have already been named so I won't repeat them.

I don't play many energy champions, but testing Zed out on the PBE, energy runes make a huge difference.

As for rage, all of them need some help. I was thinking that just giving the chalice effect to rage would help a lot, but I think the most effective choice would be rage potions. I assume they taste really bad. Give them a 30-second cooldown so that Shyvana can't just chug her way to her ult, though. If you are constantly fighting, then you're getting rage anyway.

I think Renekton needs help with his Fury.

renekton loses too much rage per second and needs to farm constantly to build it up. shyvanna doesnt have this problem, and I'm not sure about trynd, he seems to get more fury per hit and lose less fury per second, though he uses it on a different way.

Why not just have items that give mana regen also affect energy (or rage) regeneration? I mean like half the effects so that they can very well benefit from the items that would otherwise not be used by them?

Like Cauldrath suggested the Chalice, why not just make it so that Energy Champs could be granted half the regeneration benefits? Since most don't have Energy above 120 I think and rage-starved champs could get some loving, just make it so that in Dominion they get 1/3 or 1/2 the mana regeneration effects as energy or rage regeneration? That way they'd have more items to build, as well as offer more item variety for said champs (or likewise the items they already build wouldn't have wasted regen stats).

Running regen yellows on zed makes him slightly more able as an energy champ, but his ratios are pretty decent so its hard for me to say where he stands (I like him as is, and would scared of a zed who could max cdr efficiently). That said, I max w>e>q on him to optimize his energy use/fight.

Kennen is tricky. Running regen yellows on him, coupled with his passive can allow for a pretty spammy scenario with ~30% cdr. There are enough stuns happening in a team fight with his ult that he can spam q to then regen more. However, in non 4 man scenarios, he feels it from the lack of regen from constant stuns going out (which is why I run the yellows that help enough for me). Some math: Q at max rank w/40%cdr costs 21 Energy to chain cast, W costs 9, E costs 44. Upping the regen then makes it easier for him to poke while still being able to escape if needed, and more regen wouldn't really benefit him when he ults a team but would help out in 2/3v2/3's.

I haven't played lee/shen/akali enough to comment there in terms of dom, but lee has always felt like a 7ability champ to me due to his resonance mechanic.

The few downsides to this is turning out cookie cutter builds for each energy champ so that cdr/regen is optimized (if you go by an itemization route). In Zed's case, currently chain casting q costs 19energy at max rank and 40%cdr. E costs 32energy at max cdr. W can be considered virtually free by comparison. What this tells me though, is if there was no regen cap, you could push to ~33%cdr and find a sweet spot regen to pop W hit E two times during the duration, and still q chain wihtout running short on energy (and basically build last whisper + brut for damage since his base numbers with W are good).

All of that was kind of back of the envelope stuff though, but I don't think it would be too hard to figure out a cap on regen, or if there is a number that passing would be useless with all energy champs at max rank/cdr (need a dom intern this winter break??? I'm already accustomed to slave labor from my undergrad ).

Renekton, farming creeps for rage is very risky if hes not bot lane, and not having that satisfying rage empowered w is sad ;(