I also had the no sound problem with my 707. With the information in this thread, I was able to reflow Q3001 (the smaller BGA chip on the HDMI board) and my 707 is working again!
Here's what I did:
I bought a crappy Chinese SMD rework station (Hakko 850 ripoff) for like $70. I wanted one for a long time anyway:
I also got a small bottle of Kester 951 no-clean flux:
I removed the HDMI board from the chassis. It was a little tricky to figure out how to unhook the power ribbon cable that leads to the HDMI board. What I ended up doing was, I popped off the little cover on the connector, and used a sewing needle to press in each of the little spring contacts while pulling out the ribbon cable. It's tricky, but you'll see how it works after a little messing around with it.
I covered the board with aluminum foil with a little cutout for the BGA chip. This helps with keeping heat off of the other components on the board. You can find Q3001 by looking at the silkscreened labels on the PCB. The board was propped up on a couple little pieces of wood and kept perfectly level.
The rework station air temp was set to 250C and the flow rate to about the midpoint of the flow range. Don't use any of the little nozzles that come with the rework station, they're too small.
I flooded the flux under the BGA chip from all 4 sides. If you get enough in there, you can see it come out from under the other side of the chip as you tilt the board and let the flux run off.
Finally, I held the hot air wand very close to the BGA chip, perhaps 1cm to 2cm away, and moved it in a circular motion in order to evenly heat the chip. The chip was heated for 2 minutes. I could see and smell the flux burning off.
When I reassembled the unit, plugged it in, and turned it on (didn't need to connect speakers or anything), it powered up and the relays clicked and I got the speaker icons back on the display. It works like a champ now.
United Radio wanted around $400 to replace the whole HDMI board. Eff that, I fixed it myself!
Hope this info is helpful to somebody else...