It's called HDCP (Google it) Your FIOS box will not activate the component output until it gets a valid handshake on the HDMI cable. The Slingbox does not transmit a valid handshake so the the component output will work only when the TV is on and sends the right code.
The solution is to throw away the HDMI cable and feed your TV from the component OUT on the Slingbox.
You may be getting some form of SOC, which will prevent you from getting any video to your Slingbox when using an HDMI to your set-top box. There are two ways to get around this issue, which you will find in the link below. Hope this helps!
Sling Media Moderation Team
My Slingbox M1 is connected to a Verizon Fios Arris IPC1100 box with YPbPr cables, and an HDMI cable connects the FIOS box to a TV. When the TV is off, the slingbox says there is no signal. When the TV turns on, the signal appears. How do i fix this? Does the Verizon box detect that the TV is off, and therefore stop sending a video output signal?