I thought I would update this thread as I know a number of people are searching for solutions to this particular problem on Discovery 4's.
So, you have a Discovery 4 and the tailgate won't open. It will likely be one of three problems - or if you are really unlucky, all three.
It will be a problem caused by the door switch, the cable or the actuator.
Door switch - if this is the issue, when you press up on the switch, nothing happens. However, when you use the tailgate opener on the key fob, the door opens. Solution: replace the switch.
Cable - if this is the issue, when you press the switch or operate the key fob, there is 'clicking and whiring' but the door won't open, the cable is broken. This was a common problem with pre-Disco 4 cars because the cable was weaker. It was redesigned for the 4. Solution, replace the cable. See below.
Actuator - When you operate the switch or key fob, there is a weak 'whiring' sound which reduces to silence with further pushes. This is going to be the most likely cause of the problem and will likely have become progressively worse over time - in other words, the door will have previously opened but only after multiple attempts. As soon as that first happens, it's best to change the actuator because doing so with the door stuck closed is far harder than with it open.
If it is the actuator and you decide to fix it yourself, you will need part number FUG500010. This is the latest actuator. It is up to you if you wish to replace the cable at the same time. If you do, it is part number LR017470 - both parts I sourced from Land Rover Direct.
I won't attempt to explain here how to open the door or replace the parts as there are plenty of examples on Youtube. It's a bit fiddly but if I can do it, anyone can!