What is "burnout current" in an AD7730?
The "burnout currents" are current sources that you can turn on and off for diagnostic purposes.  In other words, you can use them to see if your external transducer(s) have burned out :)

The circuit looks like this:



Here, from page 25 of the datasheet: (emphasis mine)


  Burnout Currents
  The AD7730 contains two 100 nA constant current generators,
  one source current from AVDD to AIN(+) and one sink current
  from AIN(–) to AGND. The currents are switched to the selected
  analog input pair. Both currents are either on or off,
  depending on the BO bit of the Mode Register. These currents
  can be used in checking that a transducer is still operational
  before attempting to take measurements on that channel. If the
  currents are turned on, allowed flow in the transducer, a measurement
  of the input voltage on the analog input taken and the
  voltage measured is full scale, it indicates that the transducer
  has gone open-circuit. If the voltage measured is 0 V, it indicates
  that the transducer has gone short circuit. For normal operation,
  these burnout currents are turned off by writing a 0 to the BO
  bit. The current sources work over the normal absolute input
  voltage range specifications.


I don't know if this is a common feature, but I know I haven't heard of it before!