Let's first analyze the grammatical example necesse est tibi [PRO esse placidum], where PRO is often taken as the elliptical PROnominal subject of infinitival clauses. The accusative case of the subject complement of the infinitival clause (placidum) is expected to be licensed in this example since the accusative case of the elliptical subject PRO is licensed internally, i.e. no external licenser is needed for the accusative case in infinitival clauses like Necesse est [me abire] or Dicitur [te esse bonum], where the accusative case is licensed internally (see this related question for the tricky but important difference between "internal" vs. "external" licensing of the accusative case in Accusativus cum infinitivo clauses). So, to put it shortly, placidum can be claimed to receive accusative case from the elliptical subject of the infinitive (i.e. PRO), which is also internally marked with the accusative case.

Finally, your interesting example necesse est tibi esse placidus is expected to be ungrammatical since the nominative case of placidus cannot be licensed in this impersonal context: i.e. note that there is no explicit nor implicit nominative constituent in the main clause that could transfer this case to the elliptical subject of the subordinate clause and then to the subject complement placidus. In this sense it can be useful to compare this ungrammatical impersonal construction *necesse est tibi esse placidus with the grammatical non-impersonal one Homerus caecus esse traditur, where the nominative case of the subject complement in the infinitival clause (caecus) is externally licensed by the nominative case of the grammatical subject of the main clause (Homerus). Cf. also the personal construction Homerus caecus esse traditur (where the nominative case of caecus is licensed externally by Homerus) with the impersonal construction Homerum caecum esse traditur (where the accusative case of caecum (and Homerum) is licensed internally). For related descriptive discussion, see here and here, i.a.

Let's first analyze the grammatical example necesse est tibi [PRO esse placidum], where PRO is often taken as the elliptical PROnominal subject of infinitival clauses. The subject complement of the infinitival clause (placidum) can be claimed to receive accusative case from the elliptical subject of the infinitive (i.e. PRO), which is internally marked with the accusative case: i.e. no external licenser is needed for the accusative case in infinitival clauses like Necesse est [me abire] or Dicitur [te esse bonum], where the accusative case is not assigned by the matrix verb (see this related question for the tricky but important difference between "internal" vs. "external" licensing of the accusative case of the subject in Accusativus cum Infinitivo clauses). Cf. also the control structure with PRO and its dative controller (tibi) necesse est tibi [PRO esse placidum] with the more direct AcI structure necesse est [te esse placidum].

Finally, your interesting example necesse est tibi esse placidus is expected to be ungrammatical since the nominative case of placidus cannot be licensed in this impersonal context: i.e. note that there is no explicit nor implicit nominative constituent in the main clause that could transfer this case to the elliptical subject of the subordinate clause and then to the subject complement placidus. In this sense it can be useful to compare this ungrammatical impersonal construction *necesse est tibi esse placidus with the grammatical non-impersonal one Homerus caecus esse traditur, where the nominative case of the subject complement in the infinitival clause (caecus) is externally licensed by the grammatical subject of the main clause (Homerus). Cf. also the personal construction Homerus caecus esse traditur (where the nominative case of caecus is licensed externally by Homerus) with the impersonal construction Homerum caecum esse traditur (where the accusative case of caecum (and Homerum) is licensed internally). For related descriptive discussion, see here and here, i.a.

necesse est tibi esse placido (cf. Licuit esse otioso Themistocli (Cic. Tusc. 1.33)). NB: in this typical case the subject complement of the subordinate infinitival clause (placido/otioso) is assigned dative case by attraction (cf. tibi/Themistocli in the main clause).

necesse est tibi esse placido (cf. Licuit esse otioso Themistocli (Cic. Tusc. 1.33)). NB: in this typical case the subject complement of the subordinate infinitival clause (placido/otioso) is assigned dative case by attraction (cf. tibi/Themistocli in the main clause).

If you're not just interested in usage (cf. Cicero's examples above) but are also interested in knowing (how some linguists would typically explain) why your example in the title is not grammatical (in comparison with my two invented examples above, which are expected to be grammatical), here is a (typical) syntactic explanation.

Let's first analyze the grammatical example necesse est tibi [PRO esse placidum], where PRO is often taken as the elliptical PROnominal subject of infinitival clauses. The accusative case of the subject complement of the infinitival clause (placidum) is expected to be licensed in this example since the accusative case of the elliptical subject PRO is licensed internally, i.e. no external licenser is needed for the accusative case in infinitival clauses like Necesse est [me abire] or Dicitur [te esse bonum], where the accusative case is licensed internally (see this related question for the tricky but important difference between "internal" vs. "external" licensing of the accusative case in Accusativus cum infinitivo clauses). So, to put it shortly, placidum can be claimed to receive accusative case from the elliptical subject of the infinitive (i.e. PRO), which is also internally marked with the accusative case.

As for the analysis of the grammatical example necesse est tibi esse placido, the dative case of the subject complement of the infinitival clause (placido) can be claimed to be accounted for by a sort of attraction with the dative case of tibi.

Finally, your interesting example necesse est tibi esse placidus is expected to be ungrammatical since the nominative case of placidus cannot be licensed in this impersonal context: i.e. note that there is no explicit nor implicit nominative constituent in the main clause that could transfer this case to the elliptical subject of the subordinate clause and then to the subject complement placidus. In this sense it can be useful to compare this ungrammatical impersonal construction *necesse est tibi esse placidus with the grammatical non-impersonal one Homerus caecus esse traditur, where the nominative case of the subject complement in the infinitival clause (caecus) is externally licensed by the nominative case of the grammatical subject of the main clause (Homerus). Cf. also the personal construction Homerus caecus esse traditur (where the nominative case of caecus is licensed externally by Homerus) with the impersonal construction Homerum caecum esse traditur (where the accusative case of caecum (and Homerum) is licensed internally). For related descriptive discussion, see here and here, i.a.

To make things more complex (but, hopefully, more interesting for people who like syntax), note that the syntactic ill-formedness of your example in the title runs parallel to the ungrammaticality of the impersonal construction [(elliptical expletive subject) videtur esse caecus]. Note that this sequence can only be interpreted as a personal construction, i.e. [(elliptical non-expletive subject: e.g. Homerus) videtur esse caecus]. Cf. the grammatical impersonal construction [(elliptical expletive subject) videtur Homerum esse caecum].