Yes, the inverse Fourier transform is also linear just like Fourier transform. Consider the inverse Fourier transformation and Fourier transformation for f(x).
f(x) = [-∞,∞] ∫ F(k) e^{2πikx} dx
and
F(k) = [-∞,∞] ∫ f(x) e^{2πikx} dx
Since the Fourier transformation is linear, by using same augment we can show the linearity of inverse Fourier transform for F(k) and G(k).
aF(x) +bG(k) = [-∞,∞] ∫ [ af(x)+bg(x) ] e^{2πikx} dx
or,
aF(x) +bG(k) = a [-∞,∞] ∫ f(x) e^{2πikx} dx + b [-∞,∞] ∫ g(x) e^{2πikx} dx