I think Rene is referring to his Canon camera, and not the iPhone.
It looks to me like you may have your camera in Auto mode. Switch to Manual. This should lock most of the exposure settings so that you won't see the changes visible in your video. Next, meter the image first with a half press of the shutter button on the Canon while the iPhone's screen is nice and bright. The notification screen or other bright screen would work best. You basically want to make sure the camera is properly metered for a bright screen so that the results aren't blown out if you accidentally metered for a darker picture. Also make sure to have any other automatic settings on the Canon turned off. I'm not sure about your T3i, but my Nikon D7000 has an Auto ISO mode that will change ISO sensitivity depending on how the image is metered, so if you have an Auto ISO function, make sure it's disabled. This will keep camera from changing ISO and making the image darker or brighter as you switch from screen to screen on the iPhone. You may also have a continuous focus and metering mode set to On also, if the T3i has such modes. If so, make sure that is turned off as well.
As long as you're in Manual mode, the camera settings like ISO, shutter speed and aperture should remain fixed (again, unless you've got an Auto ISO mode enabled, if applicable). Get out your manual and experiment a bit if needed. I'm sure that it's possible to get the results you want without too much of a headache.
Shooting video with a DSLR does have its challenges, but what you're shooting here is very simple and straight forward. As long as you're not moving the phone to the point where the camera would need to constantly hunt for focus, you shouldn't have any problems. Just go with Manual mode and see how that works out. It should be marked on your camera dial as a big M.
Also, refrain from using P mode unless you know how it will effect the shot. P mode is kind of like "dummy" mode to some extent. It allows you to set the Program mode, and the camera is going to make all of the decisions for you to what it thinks is best; like ISO, Aperture and Shutter speeds. Looking at pictures of the T3i control dial, it doesn't look like it has as many functions as my D7000. Mine has A, S, P and M. Aperture Priority, Shutter Priority, Program Mode and Manual. I used to use P mode more in the past, especially on my older Nikon D70, but on the D7000, I usually shoot Manual probably 97% of the time, with Shutter Priority 2% and Program Mode 1%. I just like the control that Manual mode gives me, and if the picture turns out wrong, I know it was my fault and not some wonky, hard to find setting that the camera chose to use.
I hope this helps out!! Let me know how it works for you.