Here’s the difference between the two sentences:
"I'll be there by morning."
This means you will arrive sometime before or around morning. It emphasizes arrival at or before a specific time.
"I'll be there until morning."
This means you’ll already be there and will stay until morning. It implies you’re at the location now (or will be) and won’t leave until the morning.
So, if you're telling someone when you'll arrive, "I'll be there by morning" is correct. If you're telling someone how long you'll stay, "I'll be there until morning" is correct.