Inline Edit is not Available for a Specific Object

Check to see if the 'Edit' Button for the object in question has been overridden with a Visualforce page. If the Edit action has been overridden for an object, Inline Editing may not be available for that object.

To determine this:

Classic:

For standard objects: Go to Setup | Customize | <Object> | Buttons, Links, and Actions, and note the "Overridden" checkbox next to the 'Edit' standard button.
For custom objects: Go to Setup | Create | Objects | <Object>, scroll down to the Buttons, Links, and Actions section, and note the "Overridden" checkbox next to the 'Edit' standard button.

Lightning:

Setup | Object Manager | <Object> | Buttons, Links, and Actions | and then see if the Overridden column for "Edit" standard button is checked.

The Visualforce Page overriding the Edit Button must specify inlineEdit="true" within the <apex:detail>.

Note: If the Edit action has been overridden for an object, Inline Editing may not be available for that object in either the record detail, related record or (dynamic form) field section components.