It is absolutely forbidden to ascend or enter Har HaBayit in our times. The reason is that the Beit HaMikdash was built in precise areas of extreme holiness, including the Kodesh HaKodashim and other sacred zones, where entry is strictly prohibited without the proper state of ritual purity. Since today we are all considered tamei ritually impure. It is impossible to determine with certainty which areas are permitted and which are forbidden.
When visiting the Kotel, a person must approach with awe and reverence, for it is not merely a historical site but a place imbued with holiness. Our Sages teach that the Shechinah never departed from the Western Wall, and therefore it must be treated with the same respect as a Beit HaKnesset—and, in many ways, even more so. Out of this awareness, one should refrain from eating, drinking, or engaging in idle chatter in its vicinity, just as one would in a synagogue.