Yes, posting screenshots of private chats without consent can be illegal in the Philippines under several laws, though it depends on factors like the content of the screenshots (e.g., whether they contain personal data or defamatory material), how they were obtained, the intent behind sharing, and the presence of harm or malice. Below, I'll outline the key relevant laws and considerations based on legal analyses and official guidance.

Data Privacy Act of 2012 (Republic Act No. 10173)

This law protects personal information, defined as any data from which an individual's identity can be reasonably ascertained (e.g., names, contact details, or other identifiers in chat messages). Sharing screenshots without consent may constitute unauthorized processing of personal data, which includes disclosure.

When it applies: If the screenshot reveals identifiable information without a lawful basis (e.g., no consent or not for personal/family affairs), it's a violation.
Exceptions: Processing may be allowed for legitimate interests like legal proceedings, or if identifiers are redacted (making it non-personal data).
Penalties: Fines up to ₱5 million and imprisonment up to 7 years. Complaints can be filed with the National Privacy Commission (NPC), and civil damages (e.g., for moral harm) may also be sought.

Cybercrime Prevention Act of 2012 (Republic Act No. 10175)

This addresses online offenses, particularly if the shared screenshots are defamatory or intended to harm someone's reputation (cyber libel).

When it applies: Posting with malice to discredit or dishonor someone, even if the conversation was private.
Exceptions: May not apply if there's no malice, public interest, or self-defense (e.g., proving harassment).
Penalties: Imprisonment up to 12 years and fines up to ₱1 million.

Anti-Wiretapping Law (Republic Act No. 4200)

This prohibits unauthorized recording or interception of private communications, which could extend to capturing and sharing screenshots if obtained without consent (e.g., via hacking or unauthorized access).

When it applies: If the screenshot involves wire, oral, or electronic communication recorded without all parties' knowledge.
Exceptions: Does not apply to public communications or if all parties consented.
Penalties: Imprisonment from 6 months to 6 years.

Other Considerations

Constitutional Right to Privacy: The 1987 Philippine Constitution safeguards privacy of communication, providing a basis for civil claims under the Civil Code for damages like emotional distress or reputational harm.
Admissibility as Evidence: Screenshots can be used in court if authenticated, but sharing them publicly outside legal contexts risks the above violations.
Not Always Illegal: If the content is anonymized, shared privately, or falls under exemptions (e.g., for journalism or public safety), it might not violate these laws. However, the default assumption is that private chats carry an expectation of privacy.
Remedies for Victims: Affected individuals can seek a Writ of Habeas Data for privacy violations, file NPC complaints, or pursue civil/criminal cases.

Legal outcomes are case-specific, so this is not exhaustive advice. If you're dealing with a real situation, consult a Philippine lawyer or the NPC for personalized guidance.