This guide is based on Jira built-in automation functionality.

In this guide, we will transition the issue to Done when all checklist items are completed.

You can also reopen the issue in a similar way if the checklist becomes uncompleted.

Prerequisites

"Save checklist data to Jira custom fields" option must be enabled in Issue Checklist Global Settings.

Guide

Go to Automation listing page (it is either Project settings > Project automation or Jira Settings > System > Automation rules) and press Create rule button:
On the New trigger page select Field value changed option:
On the Field value changed page select Checklist Text (Field) in the first list and next select Edit issue in the second list and press Save button:
On the Add condition page select New condition option:
On the New condition page select Advanced compare condition option:
On the Compare condition page paste {{issue.properties.checklist.completed}} into the first field, select equals value from the combo list, and paste true value in the second field. Next, press Save button:

Hint: the above condition is also met when all checklist items are deleted (checklist becomes empty and therefore completed).
If you want to close the issue only when non-empty checklist becomes completed, please add one more Advanced compare condition as presented below {{issue.properties.checklist.allItems}} does not equal 0:
On the New component page select New action option:
On the New action page select Transition issue option:
On the Transition issue page select desired transition status, e.g. Done and press Save button:
On the next page, add a name/description to the automation rule and press Turn it on button:
Test your setup and enjoy

Go to any issue page, add checklist, complete the checklist (check all items) and wait a moment (Automation needs time to process the data).
Next refresh the page and notice that issue is in Done status.

Close issue when all checklist items are completed

Prerequisites

Guide