Tarasov later defended his behavior, telling Russian agency R-Sport: "It's my president. I respect him and decided to show that I'm always with him and prepared to give my support.
"What was written on that shirt was everything that I wanted to say."
Political statements are not allowed by UEFA at its competitions.
Late last year, UEFA rejected calls to separate Russia and Turkey in the draw for this summer's European Championship - and also the two countries' clubs in the Europa League draws.
Tuesday's first leg match was in the Europa League's Round of 32.
Lokomotiv lost the game 2-0 as Fenerbahçe's Josef de Souza led his team to victory with a goal in each half of the game.
Ties between Turkey and Russia have been tense since November 2015 when Turkey shot down a Russian Su-34 jet on Jan. 29 after it violated its border near Syria.