Since the furby reacts to noises, all the sensors work, but the speaker is mute. And after a while it went instantly to sleep. The speaker could have something to do with it, I advise unplugging the speaker from the circuit board and seeing if the furby will stay on and not go to sleep instantly. If this works, then the speaker has something to do with it and will need to be replaced.
Here is a guide on how to replace furby's speaker: Hasbro furby 2012 Speaker Replacement
Here is a video on how you can repair the speaker in 3 different ways: https://youtu.be/Xn5a07DFh9k
Most furby speaker specs are 32 ohm, .25 watt, 40 mm. Since you smelled the speaker burning it could have something to do with the rest of the furby not working, I advise removing it anyways since you don't want it overheating.
This video of mine may help: https://youtu.be/UteznTOUegE
You said that one of the batteries melted, but in the photos the battery terminals are clean and show no sign of battery acid or leaking. If any corrosion is found, clean it away with isopropyl alcohol and a q-tip. If the furby keeps shutting off, and if the furby is unable to move then the small 6V brushless DC motor needs replacing.
Here’s a guide that may help: Hasbro furby 2012 DC Motor Replacement
You could spin/jumpstart the motor or lubricate it. You can replace it with one from a PC DVD reader (this is the door opener motor). Shaft is smaller but it's ok, diameter is good. You need a solder iron as well to replace electronic part from DC motor.
Once you get the 6V DC motor, you’ll need to solder it on the pre-existing motor wires...I hope this helps, Thanks!