Workmen were spotted removing the flag on Sunday morning after it was reportedly hung overnight.
The stunt was branded as a "slap in the face" by local councillor Jo Broom after former Russian spy Sergei Skripal and his daughter, Yulia, were poisoned with the nerve agent 'Novichok' in March last year.
Forty-four-year-old Dawn Sturgess died after she came into contact with a perfume container used to carry the Novichok in June.
One month after the attack, Russia's Foreign Minister admitted that relations between the two countries had become "worse than the cold war".
The Dean of Salisbury, the Very Reverend Nicholas Papadopulos, said: "This was a remarkably stupid thing to do and makes light of the huge personal tragedies involved, and the damage done to the city by the unprecedented nerve agent attacks on Salisbury last year."
A Wiltshire Police spokeswoman said: "We are aware of this matter and inquiries are ongoing. It is not a police matter at the moment."