Designing high-pass filters using impulse invariance
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

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