I know that for type II FIR filter it adds a zero at $z=-1$ so it cannot design a high-pass filter.
what type of limitations the other types have (type I, III, IV)?
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
it is easy to see that
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.