This also happens to me when I attempt to open another project when I'm already in one. Essentially, in order to open a project, I have to make sure to do it from the startup screen.

Me too. Same error when I quit Unity 2017.3 (and previous versions) I am on MacBook.
Never stops for me every single time I quit Unity.
the error is:
Fatal Error
GetManagerFromContext: pointer to object of manager 'InputManager' is NULL (table index 1)

I am using Unity 2017.2.1f1 Pro and this issue still occur.
It happens everytime I try to close a project, doesn't matter if it's new just created project or some old one. The size of the project doesn't matter too. And while trying to close my project everything is saved, so no messages to save my current scene and etc.

I had the same problem.
The first time is happen when I tried to reopen a backup of a my project that was saved with unity 5.6.03f with unirty 2017.2. After first exception all projects start to made the same error.

Below the unity log file of the error.

----- Total AssetImport time: 0.155547s, AssetImport time: 0.089737s, Asset hashing: 0.000000s [0 B, 0.000000 mb/s]

Refresh: detecting if any assets need to be imported or removed ... Refresh: elapses 0.003686 seconds (Nothing changed)
[Package Manager] Server::Kill -- Server was shutdown
Cleanup mono
debugger-agent: Unable to listen on 89
[usbmuxd] Stop listen thread
[usbmuxd] Error:
[usbmuxd] Listen thread exiting
GetManagerFromContext: pointer to object of manager 'InputManager' is NULL (table index 1)

(Filename: /Users/builduser/buildslave/unity/build/Runtime/BaseClasses/ManagerContext.cpp Line: 96)

GetManagerFromContext: pointer to object of manager 'MonoManager' is NULL (table index 5)

(Filename: /Users/builduser/buildslave/unity/build/Runtime/BaseClasses/ManagerContext.cpp Line: 96)
I don't know unity in detail, but it seems to me that before close the application, the editor checks if there are some reference pointers and for a unknow reason for me, it founds some pointes that are null, so "Fatal Error."

If you open the preferences window and activate the check "Verify Saving Assets" , before shutdown unity asks a confirm to the user. This confirm in some way prevent the error and unity shutdown correctly.

Below the log file when the chek is active.
----- Total AssetImport time: 0.145808s, AssetImport time: 0.084087s, Asset hashing: 0.000000s [0 B, 0.000000 mb/s]

Refresh: detecting if any assets need to be imported or removed ... Refresh: elapses 0.003502 seconds (Nothing changed)
[Package Manager] Server::Kill -- Server was shutdown
Cleanup mono
debugger-agent: Unable to listen on 89
[usbmuxd] Stop listen thread
[usbmuxd] Error:
[usbmuxd] Listen thread exiting
Checking for leaked weakptr:
Found no leaked weakptrs.
##utp:{"type":"MemoryLeaks","allocatedMemory":162156,"memoryLabels":[{"Default":3524},{"Permanent":17064},{"Thread":232},{"Manager":328},{"Physics":8},{"Serialization":29792},{"JobScheduler":144},{"Renderer":8},{"Terrain":224},{"STL":256},{"String":32479},{"DynamicArray":31524},{"HashMap":4096},{"Curl":976},{"PoolAlloc":-224},{"ScriptManager":3272},{"Sprites":12160},{"CloudService":232},{"WebRequest":5089},{"VR":7832},{"Test":2056},{"EditorGui":252},{"EditorUtility":544},{"AssetUsage":760},{"AssetImporter":24},{"RestService":64},{"UnityConnect":9352},{"Collab":744},{"Upm":-656}]}