As pointed out by th0th, there is a breaking change in RN 0.40 for iOS. In short, RN header declarations are updated to point to the include path $(BUILT_PRODUCTS_DIR)/include/React.

To solve the issue, you have to do the following:

1. In Xcode, go to the project scheme (Product -> Scheme -> Manage Scheme -> double click your project).
2. Click on the 'Build' option at the left pane.
3. Uncheck 'Parallelize Build' under Build Options.
4. Then in Targets section, click '+' button then search for 'React'. Select it and click 'Add'.
5. 'React' should now appear under Targets section. Click and drag it to the top so that it will be the first item in the list (before your project).
6. Clean the project and build.

Note: You might still have similar header issue with other libraries (e.g. react-native-fbsdk) that are referring to those react native .h files.

In react-native 0.40

you have to replace #import "RCTBridgeModule.h" with #import <React/RCTBridgeModule.h>

then clean and build it again.

There is a breaking change on 0.40, you can see details here.

Quoting directly from the release notes:

This means that all iOS native libraries need a major version bump for RN 0.40. We attempt to minimize changes of this magnitude, and we apologize for any inconvenience caused.

So, all native iOS libraries will need an update before getting compatible with react-native version 0.40.