MFCC extraction last step DCT or IDCT
I don't believe there is a right answer.  DCT or IDCT will achieve the same purpose: decorrelating to put most energy in fewer coefficients.  Whatever you do next with your MFCC (compression, feature extraction) will work with either approach. 

Similarly, I think there are implementations that do an FFT instead of an IFFT for the first stage of the MFCC extraction.