In CGEL's terminology, we are talking about the oblique genitive construction (pp. 468-469). According to CGEL, compared to the non-oblique genitive, 'the range of semantic relations between subordinate and matrix NPs [is] considerably narrower in the oblique construction'. For example, while Mary's green eyes is equivalent to the green eyes of Mary's, we could not rephrase the summer's heat as *the heat of the summer's. (The asterisk '*' in front of something signifies that what follows is not acceptable English.)
In the case of the summer's heat, we could of course use the non-genitive of-construction the heat of the summer. But that's a separate subject.
On p. 474, CGEL provides no fewer than 23 examples of semantic relations between the genitive NP and the head. For each example, it provides a corresponding clause and a generalized schema. For example, the clause corresponding to Mary's green eyes is Mary has green eyes., and the generalized schema is [d has body part h]. Here 'd' stands for 'dependent' and 'h' for 'head', since in Mary's green eyes, eyes is the head, and Mary's a dependent. And for the summer's heat, the clause is The summer is hot., and the schema is [d has non-human property h]. The motivation for which examples to include was this (p. 473):
the semantic distinctions made are typical of those known cross-linguistically to have different structural realisations even though they are similarly expressed in English.
The list of examples is organized in such a way that the oblique construction is perfectly acceptable in the upper half, but either questionable or clearly unacceptable in the lower half (CGEL, p. 478). And the main thing that distinguishes the entries in the upper half from those in the lower half is that those in the upper half concern humans, while those in the lower half do not.
Here is the list itself:
 i Mary's green eyes Mary has green eyes. [d has body part h]
ii Mary s younger sister Mary has a younger sister. [d has kin relation h]
iii Mary's husband Mary has a husband. [d has married relation h]
iv Mary's boss Mary has a boss. [d has superior h]
v Mary's secretary Mary has a secretary. [d has subordinate h]
vi Mary's friend Mary has a friend. [d has equal h]
vii Mary's team Mary belongs to a team. [d is member of h]
viii Mary's debut Mary performs her debut. [d is performer of h]
ix Mary's book Mary writes a book. [d is creator of h]
x Mary's new house Mary owns a new house. [d is owner of h]
xi Mary's honour Mary is honourable. [d has human property h]
xii Mary's anger Mary feels angry. [d has feeling h]
xiii Mary's letter Mary receives a letter. [d is recipient of h]
xiv Mary's obituary Mary is the topic of an obituary. [d is human topic of h]
xv Mary's surgery Mary undergoes surgery. [d is undergoer of h]
xvi the room's Persian carpet The room contains a Persian carpet. [d is location of h]
xvii this year's new fashions This year is a time of new fashions. [d is time of h]
xviii the sun's rays The sun emits rays. [d is natural source of h]
xix the cathedral's spire The cathedral has a spire. [d has inherent part h]
xx the war's ancient origins The war has ancient origins. [d has cause h]
xxi the flood's consequences The flood has consequences. [d has result h]
xxii the lock's key The lock has a key. [d has associated part h]
xxiii the summer's heat The summer is hot. [d has non-human property h]
But the human-nonhuman distinction is not enough to predict whether the oblique genitive construction is possible: sometimes it is impossible even though the dependent (in the non-oblique genitive) is human. For example, note that while green eyes of Mary's is fine, *the green eyes of my boss's is not, at least to my ear. On the other hand, the green eyes of my mother's is at least marginally acceptable, again, at least to my ears. CGEL doesn't analyze this kinds of distinction (but some other source might); it seems that only some human roles, but not others, allow the oblique genitive construction.
Note also that an oblique paraphrase is normally not possible if the dependent in the non-oblique construction is an animal (the cat's owner but not *the owner of the cat's). Of course, this cat of Mary's is fine (but, again, *this cat of the owner's is not).
I should mention that, in many cases, one can use the non-genitive of-construction (either instead of or in addition to the oblique genitive), e.g. the owner of the cat. At the same time, there are certainly plenty of cases when the non-gentivie of-construction is not possible (e.g. *this cat of Mary). However, again, that's a different subject.
All this suggests that the problem with its—the reason why the genitive constructions in which its enters as a dependent cannot be turned into oblique genitives—is that its generically refers to non-human objects. More generally, the problem is that the semantic relations into which its enters as a dependent are not of the kind that can be expressible using the oblique genitive construction. While we don't have a clear necessary and sufficient condition for when the oblique construction is possible, we can say that it is at least necessary (though not sufficient) that the dependent in the non-genitive construction refer to a human—and its generically does not.
Further discussion on the semantic restrictions of the oblique genitive
In the book chapter 'The oblique genitive in English' by John Payne (in Morphosyntactic Categories and the Expression of Possession, K. Börjars, D. Denison, and A. Scott, eds, John Benjamins Publishing Company, Philadelphia, 2013), we find the following summary:
there is widespread agreement that the oblique genitive is considerably more semantically restricted than the subject-determiner genitive. Sometimes the constraint is expressed in terms of animacy. But opinions differ, e.g. the dependent must be human (Quirk et al. 1985: 1283), personal (Quirk et al. 1972: 203), or just animate (Lyons 1986; Taylor 1989). Sometimes the constraint is expressed in terms of semantic relations: Storto (2000) is most restrictive, claiming that the relationship in the oblique genitive must be one of “ownership or possession proper”. Taylor (1989) claims that the oblique genitive works best with “true” possession and various kinds of interpersonal relationship, but also “somewhat marginally” with what he calls authorship and attribution. Christianson (1997: 102) counterclaims, correctly we believe, that the authorship and attribution relations are far from marginal. For further discussion of the permitted semantic relations, see also Payne & Huddleston (2002: 478).
Examples of semantic relations occurring in the oblique genitive would be those
(7) a. that dog of yours true possession
b. that son of theirs interpersonal
c. that new book of his authorship
d. no fault of mine attribution
These relations can also be observed with the subject-determiner genitive:
(8) a. your dog true possession
b. their son interpersonal
c. his new book authorship
d. my fault attribution
However, some semantic relations which are possible for the subject-determiner genitive are claimed to be systematically excluded in the oblique genitive (examples adapted from Lyons 1986: 128–129, Christianson 1997: 101, Payne & Huddleston 2002: 478):
(9) a. *another destruction of the city’s patient of event nominal
b. *that funnel of the ship’s proper part
c. *this weather of the summer’s time
d. *that photograph of mine theme
The term “theme” would here denote the person depicted on the photograph. Compare:
(10) a. the city’s destruction patient of event nominal
b. the ship’s funnel proper part
c. this summer’s weather time
d. my photograph theme
Examples such as those in (9a–c) and (10a–c) of course involve non-human dependents, and the unacceptable examples of the oblique genitive in (9a–c) would fall under the general ban on such. However, there are clearly also some relations, like the thematic relationship in (9d), which potentially involve human dependents and in which the oblique genitive is nevertheless excluded. Similarly, there is no improvement in examples illustrating the patient relationship when the patient is human: *another destruction of the Roman legion’s appears equally unacceptable, as does the patient interpretation in an example like *another rejection of Mary’s.
The consensus therefore appears to be that the oblique genitive occurs in a subset of the semantic relations permitted to the subject-determiner genitive. The dependent in the oblique genitive must be high on the animacy scale, and not in a thematic or patient-like relation with the head noun. Syntactically, a choice of determiner other than the dependent NP itself rules out the oblique genitive. However, the definiteness effect inherent in the subject-determiner construction does not totally rule out occurrences of the oblique genitive with the definite article.