Question: How do I turn off the developer log? 

I already checked the **Output and Debugging Preferences** at the **System administration** section in the CMS but there's no option for that.

Answer: 
I've created a bug report at Ellislab and Robin Sowell replied with:

"*And yep, I’ll pass along your request re: turning it off.*"

So maybe the feature will be included in 2.6.2
I assume you mean the the notice the appears when there are notices for developers, on the control panel home screen?

AFAIK, you cannot disable the Developer Log. But these notices should only be showing to Super Admins.