Does $\triangle ABC$ exist such that $\triangle ABC \sim \triangle DEF$, with $D, E, F$ being the incentre, centroid, orthocentre of $\triangle ABC$?

Yes, this triangle exists. (Found using brute force approximation.)

Coordinates: $$A\approx(0.182,0.260)\quad B=(0,0)\quad C=(1,0)\\ D\approx(0.229,0.120)\quad E\approx(0.394,0.087)\quad F\approx(0.182,0.571)$$

Angles: $$a=d\approx107.2957\quad b=e\approx55.0744\quad c=f\approx 17.6299$$