Currently I'm using Xcode 8, Swift 3, Cocoapods (version 1.1.0.rc.2) and SQLite.swift (0.10.1). In my podfile..
pod 'SQLite.swift', :git => 'https://github.com/stephencelis/SQLite.swift.git', :branch => 'swift3-mariotaku'
Getting these two error messages:

Just tried with a simple project set up from scratch and it works there. Must be related to some other setting in your project. Can you post the full compiler error log?
I have 2 errors too, my build log is here: http://pastebin.com/38Wv2DX9
Xcode 8, swift 3
pod 'SQLite.swift', :git => 'https://github.com/stephencelis/SQLite.swift.git', :branch => 'swift3-mariotaku'

can you verify the installation path of Xcode? Very likely it is set to a non-standard path.
I solved this problem to cleaning up deriveddata folder of Xcode.
When using CocoaPods your Xcode needs to be installed in the default location.
Closing, reopen if needed
Xcode 9.4 & simulator
CocoaPod 1.5.2
use static library
The same issue when try to use use static library (remove use_frameworks!)
i unlock and modify the SQLite.h file
from
#import <SQLite/SQLite-Bridging.h>
to
#import <SQLite-Bridging.h>
it works!
Most helpful comment
i unlock and modify the SQLite.h file
from
to
it works!