A number of cards gain text through their own constant abilities:

• Maester Murenmure: During the Marshalling phase, Master Murenmure gains "…"
• Arya Stark (KL): If it is Winter, Arya Stark gains "…"
• Lion's Gate: If it is not Winter, Lion's gate gains "…"
• Jhogo (BwB): If he is attacking alone, Jhogo gains stealth.
• …

What happens if the text box of such a card is blanked? Does the gained text go away because the constant source is gone or does it stay?

As far as i understand it those arent lasting effects like a 'when revealed' plot effect or 'until the end of x phase y happens' like LOW Robb so if they get blanked the effect ends there and then.

Underworld40k said:

Indeed. They are constant effects, not lasting effects. Blanking the origin of the text gain wipes out the gained text.