What happens if an harmonic in a band-limited signal at Nyquist frequency is added to a 90 degrees out-of-phase replica of it?
No.  Adding two waves at the same frequency results in another wave of the same frequency.  Depending on the phases, there can be anywhere from complete constructive interference to maximum destructive interference.  If the amplitudes are the same there can be complete cancellation.  This applies to the Nyquist frequency too.

See the math from my answer to FFT Frequency bin relationship.

Hope this helps.

Ced