If you have the stock recovery from Fairphone Open OS 18.04 the upgrade should usually work (though some system modifications may cause problems). If your phone was encrypted in Android 5, you will need to disable secure boot before updating. If you want to flash things like openGapps, you will need to update TWRP after the update, as is discussed here.
To clarify for those finding this topic later, this is the expected behaviour. If the first boot after installing a TWRP version not released by Fairphone is not back into TWRP (but into the regular system) the update will not be persistent - i.e. it reverts to original version. The issue here, is that the reboot to TWRP isn’t working:
This is strange. It almost sounds like you’re in bootloader mode. When it stalls on the logo, does the phone respond to fastboot commands?
Which file exactly? The files ending in manual.zip will not install via TWRP. The ones ending in ota.zip may not work with some system modifications.