Without resurrecting your console, your data is lost.
Pulling the HDD and swapping it into another console will only result in a prompt on start up to restore/reformat the drive. Every drive is key coded upon formatting to work ONLY with the console in which it was formatted.
All is not lost though.
If you can work a screwdriver or have ever worked with computer hardware, there are tutorials online that outline how to disassemble your particular model PS3 and then *temporarily* fix it via the heatgun/solder reflow fix. You'll also need new thermal paste when you put it back together.
Upon successful resurrection, IMMEDIATELY do the following:
1. Manually copy any non-protected files, and game saves to an external drive. At minimum you will have these to copy to your new console.
2. If you do not have your new PS3 ready, create an archive via the back up function. This will allow you to install all non-protected files within the archive onto any PS3. All copy protected or DRM protected data will NOT transfer. This includes copy protected game saves, PSN purchases, etc.
3. If you have your new PS3 ready (this is your best option), do a direct console to console transfer of your entire HDD. You will have to deauthorize your purchased content on your old console to do this.