Will an Airbus A320 plane go into an unrecoverable nosedive and crash if a single angle of attack sensor fails, like on Lion Air JT610?
if a single angle of attack sensor fails


No.

An A320 has three AoA sensors. If one reports a different value to the other two, the avionics will ignore values from that AoA sensor and use the values from the other two.



If two of the three fail and report consistent values - the avionics will believe the third is the odd-one out and unreliable.

If those two both both report a consistent and  overly high AoA then the flight protections will apply a nose-down input in an attempt to restore the AoA to normal values and prevent a stall. this input is averaged with the pilot's inputs.

As I understand it all the following conditions must be met for the AoA protections to be erroneously invoked and cause a nose-down input:


Two AoA sensors fail and
Both report the same value and 
The value is one that would cause a stall if true.




According to Airbus' Safety-First magazine:


  In normal law, on a protected aircraft, exceeding the AOA value of the
   α PROT threshold would immediately trigger the 
  high AOA protection, thus resulting in a nose down pitch rate ordered by the 
  flight control laws. Further increasing the AOA by maintaining full back stick 
  would eventually result in reaching the α MAX threshold.





  are there any cases of an AOA sensor failing on an Airbus, and the plane not crashing?


I expect there are many.

There are even cases of two AoA sensors simultaneously failing and the aircraft continuing, after some control problems.

For example this incident  reported by Aviation Week


  German investigators studying the cause of an uncommanded pitch-down of a Lufthansa Airbus A321-200 near Pamplona, Spain, in November 2014, hope to discover the probability of similar events linked to frozen angle-of-attack (AOA) sensors as the investigation continues. In the interim, both the European Aviation Safety Agency (EASA) and Airbus have published directives alerting pilots of the issue and the workarounds.


...


  The captain continued to hold “more than 50%” rearward stick in stable flight for a period, but with help from technicians on the ground, the crew was able to reconfigure the automation into the aircraft’s alternate control law, rather than its normal “direct” law. The action removed the alpha-protection checks and canceled the nose-down input. The aircraft then continued to its destination.


...


  Airbus in early December issued a Flight Operations Transmission and Operations Engineering Bulletin to all A320-family aircraft operators noting that “if two or three AOA probes are blocked at the same angle, an increase of the Mach number may activate high-angle-of-attack protection,” which results in “continuous nose-down pitch rate that may not be stopped with backward sidestick inputs, even in the full backward position.” The recommended fix is to turn off two of the three ADRs, which puts the aircraft in alternative control law, deactivating the high AOA protection.