If the TV has HDMI input then it should be able to accept a signal from the Fire Stick, and probably from the iPhone too, but I'm not familiar with those.
Two things to try
1) If you have an existing, working, HDMI connection to the TV try moving it to a different port on the TV, just to verify that port is working.
2) If the Fire Stick is new and you've not used it yet, try using it on your home TV to verify that it works, and that you know HOW it works. The one I have is quirky. I have to turn it on and press the 'home' button and then wait several seconds before it will display anything on the TV. Until then it's just a black screen 'waiting for signal', and just about the time I think that the Fire Stick isn't working the screen will finally display.
Or, use this as your excuse to go to Best Buy and buy a spiffy new smart TV so that you don't have to connect other devices to it.