Where will your buddies sit?
Pyth, 39 38 bytes
L@+\ASzbuXGyhHyHQzpyQ&quot;It is &quot;&quot;'s turn.

This is based around repeated applications of the find and replace operation, X. The first bit defines a lookup function, y, which finds the bth player in the player order. Then, we repeatedly perform substitutions to find the final seating order, and finally print out whose turn it is.
Amusingly, the code to find the final seating order is shorter (18 bytes) than the code to print whose turn it is (21 bytes).
The code takes the seating string on the first line of STDIN, and the number of turns on the second.
Demonstration.
Explanation:
L@+\ASzbuXGyhHyHQzpyQ&quot;It is &quot;&quot;'s turn.
                                          Implicit:
                                          z = input()
                                          Q = eval(input())

L                                         def y(b): return
  +\ASz                                    &quot;A&quot; + sorted(z)
 @     b                                  (               )[b]
        u       Qz                        reduce for H in range(len(Q)),
                                          G starts as z.
         XGyhHyH                          replace in G y(H+1) with y(H).
                  pyQ&quot;It is &quot;&quot;'s turn.    Print out whose turn it is.