There are words that mean to "take out" - expunge, extricate, remove, extract, etc. But I can't find a word for "come out." In the sentence "That stain will never come out," the words come out sound very colloquial to me. Is there a single word for this?
Three words come to mind: disappear, vanish and evanish. Of these I suggest disappear and vanish as most commonly understood, while evanish fits your need well but is not well known. Vanish has overtones (below) of sudden disappearance that may not suit normal stain removal. So perhaps the safest choice is disappear
to no longer exist
(of a person or thing) to go to a place or into a condition where the person or thing cannot be seen
to disappear or stop being present or existing, especially in a sudden, surprising way
to disappear or stop existing, esp. suddenly
evanish: intransitive verb
1: Vanish, disappear
2: to cease to be
For a more formal verb, you can use dissipate, especially if you want to emphasize the time factor.
To break up and drive off
dissipate a crowd
To break up and scatter or vanish
The clouds dissipated and the sun came out. m-w
When something dissipates or when you dissipate it, it becomes less or becomes less strong until it disappears or goes away completely. Collins
According to experts at The Marble Institute, it often takes four or five applications to dissipate a stain. Old-House Journal, vols. 35, n.5
Mount immediately; otherwise, the stain will dissipate. Charles Ettensohn et al.; Development of Sea Urchins...
An oil stain on leather upholstery requires immediate attention.
"Blot up as much of the oil as you can with soft cloths or tissue," Coffey says, "then let nature take over. In most cases, the stain will dissipate as the leather gradually absorbs the remaining oil." Edward Claflin; The Experts Book of Practical Secrets