The 28-year-old Tarasov removed his team jersey at the end of the game against Fenerbahce in Europe's second-tier club competition in Istanbul on Tuesday, revealing a picture of Putin and the words: "The most polite president".
Diplomatic relations between Russia and Turkey soured after a Russian war plane was shot down by the Turkish Air Force last November.
"This is my president. I respect him and decided to show that I am ready to support him wherever I am. I wrote on this T-shirt everything that I wanted to say," Tarasov said after his side lost 2-0 in the first leg of the last-32 tie.
Lokomotiv described Tarasov's actions a "inexcusable" in a statement on their website.
"Unfortunately, the club's staff were unable to prevent this incident as the footballer had not informed anyone of his plans. What Dmitri did was inappropriate and damages the club's and his own reputation," the club said.
"Tarasov will be fined in accordance with his contract with Lokomotiv."
Russian sports news agency R-Sport quoted a UEFA source as saying that Tarasov could be banned for a minimum of 10 matches and that Lokomotiv may also face sanctions.
Igor Lebedev, State Duma Deputy and member of the Russian Football Union's executive committee, criticized Tarasov.
"You can only say thank you for such a stupid gesture," Lebedev told R-Sport. "I respect him as a footballer but this kind of action is unforgivable for a person of this level. Patriotism is not shown on T-shirts or on pictures...
"A 3-0 victory against Fenerbahce would have been the highest level of patriotism and Tarasov would have been carried out of the stadium by the fans back to Moscow," he added.
European football's governing body has taken a very serious stance against politically-motivated incidents at football matches in the past.
In July 2015, it fined Barcelona 30,000 euros ($33,438) after their Champions League final victory over Juventus during which the club's supporters displayed banners with calls for Catalonia to gain independence from Spain.
In 2013, UEFA fined Celtic 50,000 euros following a Champions League game against AC Milan at which the Glasgow club's fans unveiled a flag showing the image of Scottish national hero William Wallace and Irish Republican hunger striker Bobby Sands.
($1 = 0.8972 euros)
(Editing by Ed Osmond)
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