BEIJING—China's Internet monitors have unleashed a broad clampdown on online discussion of the 23rd anniversary of the Tiananmen Square crackdown, restricting even discussion of the nation's main stock market when the Shanghai Composite Index fell by 64.89 points—a number that made for an eerie allusion to the sensitive date: June 4, 1989.

Censors minding China's popular Twitter-like microblogging service Sina Weibo this weekend began blocking a number of terms that could refer to the 1989 Tiananmen Square crackdown, an incident...