I had the same problem, 8 mm on the left side of the screen didn’t respond. Submitted a support request on 2017-11-16, got a response within two business days with instructions to create a return request. In the meantime, I figured it would be best to order a new module and ask Fairphone to reimburse that one, instead of being without a working phone for an uncertain amount of time. So, I ordered a new module on 2017-11-23, and asked if reimbursement of the new module (depending on the assessment of the old one) was possible. Again, I received a response within two business days that it was unusual, but they would accommodate it. Next, my new module arrived 2017-11-27 (within three (!) business days after the order, pretty quick) and after a quick swap I had a well-working display again! 
So far, so good.
The return process was a bit less smooth, my first selection for pick-up by DHL didn’t work, no courier showed up, and I could not pick a new date in the portal. So, I called DHL to make a new appointment, they told me no pick-up was ever scheduled and referred me back to Fairphone to get their accountnumber first. So, I called Fairphone support. The support person understood the problem and gave me an accountnumber to give to DHL to schedule a new pick-up. So, I called DHL again and they were able to reschedule the pick-up for the same day, and this time the courier showed up. The return was delivered at Fairphone the next day. I allowed some time for the assessment, but after a week I send a message asking for an update. Within a day, I received a response that I should receive the reimbursement within five working days. It was already transferred the next day, eight working days after sending in the old display module.
Overall, I’m very happy with Fairphone support (thanks guys!), although the return portal feels a bit shaky and could use improvement. Keep in mind that I’m located in The Netherlands, which probably helps with shipping speed.
So, I would recommend the following approach:
- Make sure it’s the hardware by trying the troubleshooting steps above.
- Create a return ticket as recommended above (with order nr, IMEI, screenshot, subject, etc)
- (optional) Order a new display module
I know, this is a bit of a gamble. You might not get a refund, but a repaired screen instead (according to the Fairphone warranty). However, I reasoned I could always sell the new screen to another user or keep it as a spare. But it definitely shortens the time you are without a working phone. - Return the broken module.
If the return portal doesn’t work, call (or write) Fairphone to get their DHL account number, then DHL to schedule the pick-up. - Wait for the assessment response, write/call if it takes longer then a week.