Does a linear phase FIR filter shifted in frequency preserve linear phase?
The generalized linear phase FIR filter has the following frequency response:

$$H(\omega) = A(\omega)~e^{j (\alpha \omega + \beta)}$$ for some constant $\alpha$ and $\beta$ and $A(\omega)$ being real. The group delay of this filter is independent of frequency: $$\tau = -\frac{d (\alpha \omega + \beta) } {d \omega} = - \alpha $$

Now if you linearly shift this filters frequency response then you get the new filter's frequency response as :

$$H_2(\omega) = H(\omega-\omega_0) = A(\omega-\omega_0)~e^{j(\alpha (\omega-\omega_0) + \beta)} = B(\omega)~e^{ j ( \alpha \omega - \alpha \omega_0 + \beta}) $$

which also shows the generalized linear phase property. The new group delay is:

$$\tau  = -\frac{d (\alpha \omega - \alpha \omega_0 + \beta) } {d \omega} = - \alpha $$

the same as the previous filter. And is also independent of frequency $\omega$.

Recall to page 295 of Discrete-Time Signal Processing 2E by A.Oppenheim for a more detailed discussion of the generalized linear phase systems.