I would also contact the support. And maybe a closeby Fairphone Heaven via (List of Fairphone heavens)
Until they answer from the support I would try to test some things. Of course this is only possible in the unlikely moments when the screen lits up again. It seems to me that the problem is either the battery or the screen, maybe the proximity sensor. I once had a similar problem. Try the procedures described below and test the modules. A friend who owns a FP2 or a Fairphone angel might provide you with his/her module.
Battery: Look at the link provided in the post above (FP don't turn on ! I'm Hopeless). Maybe remove the battery, wait an hour and put it in again. If still doesn’t help, test with a working battery from a friend/angel.
Screen: Go to Settings>Maintanance>Checkup>Screen AND Drawing. Proceed both test to see if your screen works fine. If not try to test with another Fairphone screen to know if yours is faulty. If another screen doesn’t help either I assume that the problem lies in the core module. If this case occurs discuss it directly with the support.
Proximity Sensor/Top Module: My problem was that the proximity sensor was faulty and that’s why the screen stayed black. Recalibrate it via Settings>Maintanance>Proximity Sensor. If it doesn’t help, the proximity sensor might be faulty. This one lies in the top module. Again test with someone who let’s you borrow her/his module and see if it works.
If you identified a particular module that’s faulty. I would suggest to replace it.