In this case "work on" means "fix" or "repair." Without "on" the intended meaning of the phrasal verb would be lost.

'On' is necessary. Phones can be tricky to work = operate. I often work on my phone means I use my phone to do work. I don't know how to work out new microwave means I don't know how to operate it.

I do not understand the sentence. Specifically, it is not clear to me what is being "worked on". Is it the phone itself? Or is it some unrelated "issue"? After all, people talk on phones to discuss all sorts of things unrelated to phones. Without further context, I cannot suggest anything.

Why not remove 'work on' and replace with 'use'?

Certainly! Here's a revised version of the sentence without the word "on": "Phones are great for providing immediate support for simple and easy-to-fix issues but can be tricky to work with when cases become more complex."