Unfortunately, no. iOS does not support mirroring to a Roku device. With the Roku app on your iPhone, it is possible to play music and videos that are on the phone itself on a Roku device, but not from other apps on the phone. Android devices generally can screen mirror well, and Windows 10 laptops appear to work as well. My Windows 10 desktop doesn't show any networked devices for casting, but I think that's because it doesn't have a wireless network adapter, and I think that's a Windows requirement for screen casting to other devices.
Dan Roku Stick (3600), Ultra (4640), Ultra (4670), Premiere (3920), Insignia 720p Roku TV, Sharp 4K Roku TV, Nvidia Shield, Windows 10 Pro x64 running Serviio and Plex on a wired Gigabit network.