This is what happens to him in the canon story. 

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"Ma'am.." A doctor said quietly and walked over to me. "Yes..?" I asked tiredly. I had spent all day worrying about my baby. My poor baby.. He had been diagnosed with lung cancer and now it was getting worse. 

"We're sorry, but.. I don't think.. we can save him.." The doctor told be gently. I'd been ready to hear that, or I thought I was. I teared up and tried to calm myself down. "I-.. can I.." I stammered, not able to finish my sentence. 

The doctor nodded and led me into the room. "Hey Bugga Boo!" I said, gently. Greed looked at me, still laying in his bed. "Hey mama..!" He greeted hoarsely, "What are you doing here..?". "I've come to say goodbye.." I said. 

"Where are you going..?" Greed asked. I struggled to stay smiling. "Ah.. just on a little trip.." I mumbled and took a seat next to the hospital bed. "Mama..? Will I ever see you again..?" Greed asked gently. 

I gently took his hand and ran my thumb across his knuckle. "S-sure... sure you will, kid.." I told him, tearing up, yet somehow managing a smile. "You know goodbyes are never forever.." I said. "Well.. then.. I'll see you later, Mama..! I love you.." Greed said, his voice softening and his breath slowing. 

I couldn't hold it in. I cried and cried. "I-i-i lo-love you t-too bugga boo.." I said gently, then squeezed his hand one last time.