Pulmonary tuberculosis (PTB) is a bacterial infection that affects the lung. PTB remains one of the infectious diseases with the highest global mortalities. Chest radiography is a technique that is often employed in the diagnosis of PTB. Radiologists identify the severity and stage of PTB by inspecting radiographic features in the patient's chest X-ray (CXR). The most common radiographic features seen on CXRs include cavitation, consolidation, masses, pleural effusion, calcification, and nodules. Identifying these CXR features will help physicians in diagnosing a patient. However, identifying these radiographic features for intricate disorders is challenging, and the accuracy depends on the radiologist's experience and level of expertise. So, researchers have proposed deep learning (DL) techniques to detect and mark areas of tuberculosis infection in CXRs. DL models have been proposed in the literature because of their inherent capacity to detect diseases and segment the manifestation regions from medical images. However, fully supervised semantic segmentation requires several pixel-by-pixel labeled images. The annotation of such a large amount of data by trained physicians has some challenges. First, the annotation requires a significant amount of time. Second, the cost of hiring trained physicians is expensive. In addition, the subjectivity of medical data poses a difficulty in having standardized annotation. As a result, there is increasing interest in weak localization techniques. Therefore, in this review, we identify methods employed in the weakly supervised segmentation and localization of radiographic manifestations of pulmonary tuberculosis from chest X-rays. First, we identify the most commonly used public chest X-ray datasets for tuberculosis identification. Following that, we discuss the approaches for weakly localizing tuberculosis radiographic manifestations in chest X-rays. The weakly supervised localization of PTB can highlight the region of the chest X-ray image that contributed the most to the DL model's classification output and help pinpoint the diseased area. Finally, we discuss the limitations and challenges of weakly supervised techniques in localizing TB manifestations regions in chest X-ray images.