Hello,
I would like to send push notifications to my mobile phone or desktop app, with a reminder of a task due date approaching. Is this currently possible?
Welcome to the Asana Community Forum @Chrimata_Group 
Yes that is possible, have a look here: Getting started with the Asana iOS app | Product guide • Asana Product Guide
- A task you’ve been assigned is due (sent around midnight on the day of the task’s due date)
And for desktop notifications see here: Asana app not showing in the notification and action setting Win 10 - #2 by Rebecca_McGrath