Formal:
* Rire à gorge déployée: This literally translates to "laugh with your throat deployed" and is the most formal way to express a hearty laugh.
* Rire aux éclats: This translates to "laugh in bursts" and is another formal way to describe a loud, joyful laugh.
Informal:
* Rire à s'en tordre les côtes: This means "laugh until your ribs crack" and is a very common way to say "laugh your heart out" in informal settings.
* Rire aux larmes: This means "laugh until you cry" and is another good choice for informal situations.
* Rire comme un fou/une folle: This literally means "laugh like a madman/woman" and is a playful way to say "laugh your heart out".
Idiom:
* Se tordre de rire: This means "to twist with laughter" and is a very expressive way to say "laugh your heart out."
The best translation will depend on the context and the tone you want to convey.