How come HDMI won't work on my Poloroid flat panel TV?

Not that I heard of. You don't specify what MBP model you have. I'll assume below you have a mini-displayport or thunderbolt (same as mini-displayport for this application) connection. If not you need a appropriate adapter to convert your (dvi?) video out to hdmi.

Set the TV to the input you connected the HDMI cable to.

On your computer, if you open the Displays preferences and do not see an Arrangement tab (or there is an Arrangement tab that shows only a single monitor) then there is something physically wrong with your connection. Try clicking Detect Displays just to see if the computer can now sense the presence of the connection. Check your cables and make sure the mini-displayport plug is pushed firmly in so that essentially all the metal part disappears inside the socket. Mini-displayport plugs can be a little tricky in that you need to make sure the plug is fully seated in their socket.

If you do have an Arrangement tab, and it shows two screens, then click Gather Windows to bring the TV's Displays preferences on to your main screen if the TV is not yet showing an image. Set the proper resolution (probably 720p, 1080i, or 1080p). Set the refresh rate (probably 60Hz) if that preference is shown. You should see a picture at this point. I recommend turning off mirrored displays if it is enabled (more about this below).

You may have to play with the underscan/overscan settings on your TV and the displays prefs for the TV if the picture is too small or larger for the TV screen.

Also, you need to set your Sound preferences to output to the HDMI connection. If the MBP doesn't support audio through the mini-displayport then you will need to connect your stereo audio out to your TV (3.5 mm stereo jack for the compter to usually RCA jacks on the TV).

And when running Lion, do not run in Lion full screen mode.

Is there an Arrangment tab or not?

Then your computer is detecting the TV.

After doing the Detect Displays and Gather Windows I would think you could set the proper resolution for that TV. I would expect 720p, 1080i, or 1080p but maybe there's a 1920 x 1080 or 1280x720. Is there a refresh rate? Probably need 60Hz.

There are versions of HDMI (see Wiki). I suppose there is the possibility that your TV is an older version not supported by your computer (or adapter?). I'm not sure about downward compatibility of HDMI like, for example, usb3 being downward compatible with usb2.

Also the TV is expected to send the proper EDID information to your computer. It is this information Displays preferences need to show you the acceptible resolutions and refresh rates.

Finally there's good 'ol HDCP (copy protection). I'm not sure how that might come into play here but if the entertainment industry can find a way to screw up our own private equipment that we paid for and own and want to use the way we desire they will.😢