I've had burnout a few times in my career. I think it happens to anyone who really enjoys their job. It's like being placed in front of a pile of candy -- just one more piece!
But burnout is an awful feeling. You aren't able to work. For me, I would stare at the screen for hours on end without being able to get started.
I found that 2 things helped me the most. First the pomodoro system. You are probably familiar with it, but just in case: you have a 25 minute session, where you devote yourself to nothing but concentrating on work. For me, it's unbelievably important to get my first pomodoro in as early in the day as possible. I also found that it is just as important that this pomodoro is coding: not reading mail. Not doing a status update. Not having a standup meeting. Not investigating some bug. I've got to write code as soon as I can in the day. That flips a switch for me. Very important, especially if you are burned out: show up for work significantly before your fist meeting of the day -- the meeting will suck the life out of you and you need some life to suck.
The next thing, strangely, is a TODO/log of what I'm doing. Being burned out means that everything is a barrier. Usually the barrier is too high, so you are stuck staring at it with no idea how to get over it (because normally you would just step over it).
When I first start a pomodoro, I write a TODO of what I'm going to do for the next 25 minutes. I find it's best to spend as little time as possible doing it. Just write down the first thing that comes to your mind about what you need to do. Write 3-4 items at least. Then start tackling the first one. Inevitably, you will find that your TODO was not good -- because you didn't spend time thinking about it. As soon as you realise what you need to do, update your TODO and then do it.
It's tempting to write the code and then write down what you did. I find that it works best to do it the other way around even though it breaks the flow. When you are burned out, flow is quite hard to maintain and any distraction is going to grab you hard. By focussing on making sure your TODOs are correct, it means that you can easily jump back in.
The biggest thing to realise is that you will often be punted out of the zone and will be staring at your screen. What you need to do is to say to yourself, "Just one more" -- which means, making an easy TODO for yourself and writing the code to fullfil it. Then "Just one more". Eventually you will get distracted. When you finish being distracted, turn on the pomodoro timer and say, "Just one more".
At first you may find it hard to do more than 4 pomodoros in a whole day. Don't worry about it. Just try to do as many as you can. Then the next day try to do more than the previous day. I have found that I can claw my way out of burnout pretty quickly like this. It doesn't take long before I'm enjoying my work again. After that, the productivity starts coming back and things start to feel easy again.
Hope this helps.