Though, I can come up with a pretty convincing argument for the opposite.

Diseases only become drug-resistant as a result of natural selection in an environment in which drugs which try to treat the disease are used.

Third world countries have issues with distributing drugs/treatments to everyone in the society, and so it is likely that diseases will not be completely eradicated, but instead exist in an environment with drugs in use. Even in individuals, there are problems with consistently treating the disease, and so it's likely to pressure the disease without curing it.

On the other hand, diseases rarely become drug-resistant when they're not exposed to the drugs.

Therefore, treating people in third-world countries increases the probability of producing drug-resistant strains of existing diseases.