You can split the HDMI output of the TIVO and send it to as many TVs as the distribution amplifier is designed for. You won't be able to watch different channels at the same time though. You also want the distribution amp to carry the IR remote control signal so you can operate the TIVO from the remote location. If they have an app for that then you can use that instead.
You can run HDMI cables or with the correct distribution amp use CAT5e or CAT6. This requires adapters at each TV to convert back to HDMI but the wire is much cheaper and easier to run.
If you do this before permanently running the cables I would try the set up to see if it works. HDMI can be tricky.