There is no evidence that transcendental meditation confers any specific benefit separate from the benefits achieved by relaxation. (ref - sciencebasedmedicine.org, and ref - American Heart Association Scientific Statement).
From Dr. Steven Novella:
For example, there are many studies showing that transcendental meditation is effective for lowering blood pressure. However, studies generally compare TM to no intervention, not to other forms of relaxation. The parsimonious interpretation is that TM confers the generic benefits of relaxation, but there is no evidence to suggest it confers any specific benefits.
From the AHA's Beyond Medications and Diet: Alternative Approaches to Lowering Blood Pressure (at p. 6):
The overall evidence supports that TM modestly lowers BP.
It is not certain whether it is truly superior to other meditation
techniques in terms of BP lowering because there are few
Given that no specific effects above relaxation have been demonstrated in individuals practicing TM, I don't see how there could be specific effects in entire populations that interact with those individuals.
However, this research (above) is focusing on clinical outcomes of transcendental meditation. The Maharishi effect theory makes claims that are more vague. It claims that there is a "collective consciousness" or a "unified field" of consciousness, and that the effect on this field by the TM practitioners can result in TM-coherence even in non-practitioners, among other things.
Here is a page that examines the specific evidence that proponents of the Maharishi effect point to.
One specific peer-reviewed critique of the Maharichi effect (Evaluating Heterodox Theories, text available here) says the following:
MT [Maharichi effect theory] has serious problems regarding the clarity and integrity of its arguments, and it does not cohere well with other strongly confirmed theories, hence conflicting with the evidence supporting those theories. MT is under-articulated, often vague or enigmatic, reliant upon specious analyses, and silent about key processes that link causes to their alleged effects. These defects are not uncommon in novel theories, but in this case they allow nothing better than crude plausibility arguments for its extraordinary predictions.
It goes into detail about what the Maharichi effect theory actually claims, the purported evidence supporting its predictions, and gives alternative explanations for that evidence that have not been ruled out by the theory's proponents.