I found that after installing the Roku Premiere+, it does the same thing.
I started swapping HDMI cables. I found that if I run the Roku P+ directly to my TV, the problem doesn't happen. It only seems to happen when I run the Roku P+ (or 3) through my Denon AVR 3313CI receiver.
I went through the options on the Roku P+ and found one named "1-Touch Play" which was selected and said something like "Tell the TV to switch to this input any time I hit a button on the remote". Sure sounds like my problem. I turned that off and the problem went away.
I put the Roku 3 back into the system to see if that was the problem with that device as well. After turning that feature off, it started working fine.
My assumption at this point is that something on my TV broke such that this feature which used to work senselessly, now causes the TV to blank out. My TV is pretty old (it's a 2009 model Mitsubishi WD-65737). I could try doing a factory reset on the TV, but honestly, turning the feature off works for me (I wouldn't want it on anyway) so I'm happy.
Thanks for all the help.