I connected my new Sony Blu Ray player to my old Hitachi TV using a HDMI conversion cord (TV doesn't have HDMI, so we got a cord converter). My screen blinks between the Blu Ray menu and black and the

SOURCE: 42LG50 TV - Intermittant sound problems with blu-ray HDMI input

You say this doesn't happen on regular cable or when playing a regular DVD. Hmm. Is this Blu-Ray player connected by simplink? HDMI is a bi-directional format cable, so  there are cases in setups where your tv can send commands to your blu-ray player and vice versa. It could be a problem with your blu-ray player that it is sending for some odd reason a mute command to your tv. Or is processing a command that your TV interprets to be a mute command when playing blu-rays. Because if it was the TV itself chances are you'd see this when watching a normal DVD or on cable as well. Since you say you're not, it leads me to believe the issue lies with the blu-ray player itself. 

SOURCE: When trying to connect my 4695D to a new Sony Blu

The problem has been solved. A previous post to this site was having a similar problem hooking up a Playstation Blu Ray player to the same TV.

A solution was posted to that problem and I followed the same instructions.

It has to do with the set up porgram on the Blu Ray component. You need to select Settings then 108024Bit needs to be set to off.

SOURCE: My HDMI connection does not work with Blu Ray DVD

After reading your post a couple of times, it just seems your are in a little of a pickle here. The only thing I can suggest, since your tv will work with fios and not the your BD player(I assuming your using the same port on the tv with the same hdmi cable), is all about firmware. Maybe your tv is not compatible with the the BD player. Try updating the firmware on the tv.

SOURCE: Plasma tv showing "Mode not supported" message

YOU HDMI SWITCHER IS DEFECT OR YOUR TV IS NOT COMPATIBLE

SOURCE: Westinghouse LCD TV 32'' not working with Blu Ray DVD player

I have the same issue with this TV and an Xbox. it can't keep a HDCP connection and my research shows that it either can't support HDCP or I have a cable that isn't able to support the speed (It pretty cheap) or signal, or it's a bad cable. I'm going to swap it out to see. But Blue Rays use HDCP and you may be having the same issue.