Designing high-pass filters using impulse invariance One of the main disadvantages of realizing digital filters using impulse invariance is aliasing. According to the Nyquist sampling criterion, in order for the frequency response of the digital filter to match that of the corresponding analog filter with impulse response $h(t)$, $h(t)$ has to be band-limited. That is, the analog filter has to be low-pass. Does this mean we cannot map high-pass analog filters to high-pass digital filters using impulse invariance? Or is it just a matter of mapping a low-pass analog filter to a low-pass digital filter with frequency response $H(e^{j\Omega})$, and then finding the frequency response of the corresponding high-pass filter simply by substracting it from 1?
You're right, you can't use the impulse invariance method for transforming continuous-time filters with a transfer function that is not strictly proper, i.e., for which the degree of the numerator is not strictly smaller than the degree of the denominator. High-pass and band-stop filters have transfer functions with numerator and denominator polynomials of the same degree, which means that the corresponding partial fraction expansion has a constant term. The latter corresponds to a (scaled) Dirac delta impulse in the impulse response, which obviously cannot be sampled.

Take a first-order high-pass filter as an example:

$$H(s)=\frac{s}{s+a}=1-\frac{a}{s+a}\Longleftrightarrow h(t)=\delta(t)-ae^{-at}u(t)\tag{1}$$

Now you can't determine $h(nT)$, $n\in\mathbb{Z}$, at $n=0$.

In sum, for the impulse invariance method to be applicable we require

$$\lim_{s\to\infty}H(s)=0\tag{2}$$