The signs are there because the law requires them to be. Federal law requires that the US government periodically assess the aviation security practices of foreign airports. If they are found to be deficient, the law requires that notices be posted:
(d)Actions When Not Maintaining and Carrying Out Effective Security
Measures.— (1) When the Secretary of Transportation decides under this
section that an airport does not maintain and carry out effective
security measures— (A) the Secretary of Transportation shall— (i)
publish the identity of the airport in the Federal Register; (ii) have
the identity of the airport posted and displayed prominently at all
United States airports at which scheduled air carrier operations are
provided regularly; and (iii) notify the news media of the identity of
Airlines are also required to provide similar notice to people who book tickets to the airport.
Back in December, they announced that MNL airport had failed the assessment (there's been some history of problems there) and that advisories would be published. Hence the signs: the law says notices need to be posted, so here they are.
The law does permit a suspension of flights under such circumstances, but that's a discretionary act, not something that would be required, and I'm not aware of any public statements indicating that anybody is proposing such a measure. The authorities in the Philippines are trying to take steps to rectify the problems, with some financial support from the US government. So yes, it is meant as a warning to force them to improve their security practices and a warning to passengers that they could reconsider their travel, though there's at least the theoretical potential the US government could suspend flights if the airport doesn't improve.
These situations can often be addressed if the airline hires additional security (and expensive security consultants, naturally) to screen only US-bound flights. In some rare cases, airlines will build in a stopover where passengers can be rescreened before flying to the US. Until recently, this was done by Kuwait Airlines, where flights to the US made a stop in Shannon until the US authorities were satisfied with Kuwait's airport security improvements. So there may also be other options beyond a suspension of flights in some cases.