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)?
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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.