Jelly, 9 bytes

rµJÆḞ*@ḅ1

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Jelly, 9 8 bytes

rJÆḞ*@rS

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-1 byte thanks to ZippyMagician

How it works

rJÆḞ*@rS - Main link. Takes a on the left and b on the right
r        - Range from a to b
 J       - Length range of that
  ÆḞ     - n'th Fibonacci number of each
      r  - Range from a to b
    *@   - Raise each to the power of the Fib numbers
       S - Sum