Why does the FAA mandate a temperature indicator for all manned balloons?
A hot air balloon (including the air in the envelope) has tremendous mass, so it takes a long time for a change in buoyancy to cause a change in vertical speed.  Watching for temperature changes on the Envelope Temperature Indicator will help a pilot anticipate changes in the balloon's vertical speed before they have actually had time to occur.   

From page 7-4 of the FAA's Balloon Flying Handbook: 


  "Ascents and Descents
  
  The temperature of the air inside the envelope controls balloon
  altitude. A balloon that is neither ascending nor descending is in
  equilibrium. To cause the balloon to ascend, increase the temperature
  of the air inside the envelope. If the temperature is increased just a
  little, the balloon seeks an altitude only a little higher and/or
  climb at a very slow rate. If the temperature is increased
  significantly, the balloon seeks a much higher altitude and/or climbs
  faster. If the balloon is allowed to cool or hot air is vented, the
  balloon descends.
  
  Even without the input action by the pilot, it must be remembered that
  the air inside the envelope is dynamic. The air mass is constantly
  moving within the confines of the envelope, attempting to seek a level
  of equalization. While it varies with each envelope, input action by
  the balloon pilot can take from 6 to 15 seconds to be realized as a
  reaction by the balloon. Planning the maneuver, anticipating the
  reaction time, inputting the proper burn, and observing the reaction
  must result in smooth and natural movement by the pilot."


Also, as another related answer has noted, under some conditions it is possible for a pilot to bring the envelope to an unsafe temperature by excessive operation of the burner.  Monitoring the Envelope Temperature Indicator allows this to be avoided.  For a given desired maximum envelope temperature, the outside air temperature determines the total gross lift that is available at any given density altitude, or the density altitude that can be reached with a given gross weight.  See the FAA's Balloon Flying Handbook for more.