To address this question, linear interpolation is NOT a linear phase filter except for when we are interpolating to half a sample. The necessity for a linear phase filter is that the filter coefficients be complex conjugate symmetric or anti-symmetric. Linear interpolation as an FIR filter is done with two coefficients $a$ and $b$ such that:
$$y[n] = a x[n] + b x[n-1]$$
with $a+b = 1$
and the fractional interpolation in samples given by $a/b$
Given the first statement on what makes a filter linear phase, this can only occur when $a= b= 0.5$. For any other values of $a$ and $b$, the result will not be linear phase.
The reference to interpolating filters in Dr. Rice's book are likely referring to higher order polynomial interpolation and not "linear interpolation"; using a multitap filter with more than 2 coefficients rather than the "linear interpolation" between two samples that the OP is considering (given the comment under the originating question). Higher order polynomial interpolation can and typically do have linear phase given an implementation with symmetric coefficients. This is the typical but not necessary condition as we can make interpolators that are not linear phase as well. Consider a typical interpolation process of doing a zero-insert followed by filtering to remove the images the zero-insert creates. Certainly a linear phase filter would be preferred to for the filter to minimize distortion and be closest to the ideal Sinc function interpolator, but this is not necessary to achieve interpolation within target requirements. Any realizable interpolation approach linear phase or otherwise will have a finite distortion regardless since perfect interpolation requires an infinitely long filter.