What are the limitations of FIR filters?
Note that this is not about general limitations of FIR filters, but about the special case of linear-phase FIR filters. If you understand why a type-II linear-phase FIR filter has a zero at $z=-1$, then the limitations of the other types should be obvious too.

It's always about zeros at either $z=1$ (DC) or $z=-1$ (Nyquist). Given the transfer function

$$H(z)=\sum_nh[n]z^{-n}\tag{1}$$

it is easy to see that

$$H(1)=\sum_nh[n]\tag{2}$$

and

$$H(-1)=\sum_n(-1)^nh[n]\tag{3}$$

Now look at examples of the four linear-phase FIR filter types. An example of an impulse response of a type-I filter is

h = [1 -1 2 3 2 -1 1]

Using $(2)$ and $(3)$ can you say anything about zeros at $z=1$ or $z=-1$?

An example of a type-II filter is

h = [1 -1 2 2 -1 1]

Clearly, applying Eq. $(3)$ results in a zero at $z=-1$, regardless of the actual values of the impulse response.

The following two filters are type-III and type-IV filters, respectively:

h = [1 -1 2 0 -2 1 -1]

h = [1 -1 2 -2 1 -1]

Now apply Eqs $(2)$ and $(3)$ and see what you get.

This answer contains more information on the four types of linear-phase FIR filters.