If you are a resource level user a task cannot be closed unless you are the owner of the task. The reason it works this way is Intervals enforces a workflow that allows a task to be closed only after the owner has verified the task. As a resource level user if you are BOTH the owner and the assignee the task can be closed. A common workflow is to update the task and setting it to a task status of “Verify & Close” to notify the task owner that the work is done. Then the task owner can review the work and reopen the task or close it.
Manager and administrator level users can close tasks regardless of who owns the task. Here is a review of the four user levels with videos.