Can a causal filter without phase shifts exist?
Suppose that a linear filter has impulse response $h(t)$ and frequency response/transfer function $H(f) = \mathcal F [h(t)]$, where $H(f)$ has the property that $H(-f) = H^*(f)$ (conjugacy constraint). 

Now, the response of this
filter to complex exponential input $x(t) = e^{j2\pi f t}$ is
$$y(t) = H(f)e^{j2\pi f t} = |H(f)|e^{j(2\pi f t + \angle H(f))}$$ and if
we want this filter to cause no phase shift, it must be that $\angle H(f) = 0$
for all $f$.  

How about if, instead of no phase shift, we are willing to allow
a fixed constant phase shift for all frequencies? That is, $\angle H(f) = \theta$ for all $f$ is acceptable to us where $\theta$ need not be $0$? The extra 
latitude does not help very much, because $\angle H(-f) = -\angle H(f)$, and
so $\angle H(f)$ cannot have fixed constant value for all $f$ unless that value
is $0$.  

We conclude that if a filter does not change the phase at all, then
$H(f)$ is a real-valued function, and because of the conjugacy constraint, it is
also an even function of $f$.  But then its Fourier transform $h(t)$ is a
an even function of time, and thus the filter cannot be causal (except in trivial cases): if its impulse
response is nonzero for any particular $t &gt; 0$, then it is also nonzero for $-t$ (where $-t &lt; 0$). 

Note that the filter need not be doing any
frequency suppression, that is, we did not need the assumption that
some frequencies are "removed" by the filter (as the OP's
professor's filter does) to prove the claim that zero phase shift
is not possible with a causal filter, frequency suppressor or not.