Is there a threshold voltage below which batteries are definetly dead and has to be recycled?
It's more than likely that the protection circuity has shut off all access to the battery due to it being over-discharged. 

Li Ion is an inherently dangerous battery chemistry and lots of precautions are made to try and make it safe, one of these is effectively a fuse in the protection circuity that when over-discharge of the battery occurs effectively bricks the battery - without diverting the protection board at least (tip: don't do this). 

The problem is that in an overly discharged Li Ion, tiny needle-like crystals begin to form in the liquid electrolyte. These crystals can eventually bridge the anode &amp; cathode causing an internal short in the battery, usually leading to fire and/or an explosion. Here's a video demonstrating what happens to a shorted lithium ion battery.

Even though the battery is showing 0v there probably is still some charge left in the battery, but it has dropped below a voltage at which the battery can now be considered "unsafe" and the protection circuitry has broken the connection between the battery terminals and the terminals exposed for connections (probably where you are testing). In other words, the terminals you are using to test the voltage of the battery are probably not the terminals of the actual battery itself, there is probably a tripped fuse somewhere between the actual terminal and the point at which you are testing.

For reference, according to The Journal of Chemical Thermodynamics (referenced on this Wikipedia page)


  Lithium-ion cells are susceptible to damage outside the allowed
  voltage range that is typically within (2.5 to 3.65) V for most LFP
  cells. Exceeding this voltage range, even by small voltages
  (millivolts) results in premature aging of the cells and, furthermore,
  results in safety risks due to the reactive components in the cells.


As @BruceAbbott pointed out in the comments the quote above specifically talks about Lithium Iron Phosphate (LiFePO4)batteries that have a slightly different safe range than standard Lithium Ion. For LiIon the range is typically 2.7-4.2V.

NiMh and alkaline batteries do not suffer from the crystal growth issue, as far as I am aware.