(יב) מַכֵּ֥ה אִ֛ישׁ וָמֵ֖ת מ֥וֹת יוּמָֽת׃ (יג) וַאֲשֶׁר֙ לֹ֣א צָדָ֔ה וְהָאֱלֹהִ֖ים אִנָּ֣ה לְיָד֑וֹ וְשַׂמְתִּ֤י לְךָ֙ מָק֔וֹם אֲשֶׁ֥ר יָנ֖וּס שָֽׁמָּה׃ (ס) (יד) וְכִֽי־יָזִ֥ד אִ֛ישׁ עַל־רֵעֵ֖הוּ לְהָרְג֣וֹ בְעָרְמָ֑ה מֵעִ֣ם מִזְבְּחִ֔י תִּקָּחֶ֖נּוּ לָמֽוּת׃ (ס)
(12) He who fatally strikes a man shall be put to death. (13) If he did not do it by design, but it came about by an act of God, I will assign you a place to which he can flee. (14) When a man schemes against another and kills him treacherously, you shall take him from My very altar to be put to death.
See Rambam's laws Hilchot Rotzeach u'Shmirat Nefesh- Chapter Five to gain clarity as to when the altar saves and when it does NOT save (excerpt below).
If someone takes refuge on the altar, he is not left there. Instead, he is given guards and taken to a city of refuge.
When does the above apply? When one is obligated to be exiled. If, however, a person feared that a king will have him executed as is the king's authority, or that the court will execute him as an immediate directive, and fled to the altar and held on to it, he should be saved.
This applies even if he is a commoner. He should not be taken from the altar to die unless he was sentenced to death because of the testimony of witnesses who delivered a warning, as is always required with regard to those executed by the court.