Is it permissible for one of the worshipers to wait for the fourth rak'ah in congregational prayer to open the door for those who are late, in order to avoid closing the mosque and losing the place of prayer?

There is no harm in a person delaying to monitor the door if a benefit results from it, such as preventing a predicament or being deprived of performing prayer in the mosque. If he joins the Imam before the final bow of the last rak'ah, he has caught the congregation without dispute. If he catches up with him after the bow and before the tasleem, then he has caught it according to some scholars.