Can you wear heavy armor and still sneak successfully? If so what perks are needed?

Yes you can you just need perks in the sneak trees and heavy armour tree.

Yup, you can. Both Heavy armor and sneak have perks to muffle you, and make your armor weigh nothing. There's even a psecial set of armor for you: The Ebony Mail. Adds a sneak bonus, gives you a poison cloud and makes you temporarily invisable when you sneak.

I put perks into heavy armour but none into sneaking and I have no problems you just need to get your skill high enough and your golden.

You can do it at low levels with The Steed sign. It's arguably the best sign in the game: +100 Carrying Capacity, and any worn armor weighs nothing. In other words, you get not only +100 inventory space, but you also get the additional space freed up by nullifying the weight of your armor set. And the zero weight means your armor has no speed penalty, no stamina penalty, and no noise penalty. It's the equivalent of running around naked, only you get the defense stats (and appearance, obviously) of your armor. Once you get to a higher level, you can pick up Conditioning (level 70 Heavy Armor perk) which has the same zero weight effect as The Steed.

I'm currently working on a Dunmer Vampire focusing on Sneak and Illusion, using Heavy Armor with a dagger (right hand) and destruction spell (left hand) for combat, and it's working very well so far. It's a great blend of all 3 playing styles (Illusion, Destruction, and occasionally Restoration for magic; One Handed and Heavy Armor for melee; Sneak and occasionally Alchemy for stealth). Illusion is very useful for Sneak as well; you can use Muffle as JerZeyCJ suggested, as well as Invisibility, Calm, and Frenzy, all of which are very helpful for stealth-based characters.

For all of you that put perks into Light Armor, and want to wear Heavy Armor there is always the full set of Falmer Armor. It uses the Light Armor perk tree.