Is there a way that a Task can send you an email or Pop Up that an Item is coming due?
If you have a step in your workflow where you would like JIRA to send an email notification, you may be able to rig something if you have JIRA Suite Utilities and/or ScriptRunner Plug-ins. (Im not sure which does what since we installed them both before our initial rollout) Id have to research further how - and it would depend on your configuration- but Im pretty confident it is possible with these plugins.