The remote control capabilities of a Slingbox do not control a TV, they control a cable/satellite box.
The reason your TiVo remote could control your TV's "Input" is because the TiVo remote is likely a universal remote, where you entered codes to teach the remote how to control your TV. If you're trying to change the input on the TV where the Fire TV is attached, you'll need to use the remote that came with the TV or a universal remote that's been programmed to support your TV (such as the TiVo remote you mentioned).
I have a sling box and tivo. I recently purchased an amazon fire tv and I am ordering a new sling player so they will be compatible and i am getting the sling app. Is there any problem changing the television input via the remote with the new sling player? I just tried to do it from my house with sling.com or the sling player app on my iphone and can't do it although I could do it directly with the tivo remote.