The Foxes welcomed Luciano Spalletti's side to the King Power Stadium in the opening group game, and Ayoze Perez gave them an early lead with a close-range finish.
Barnes, who put in the cross for Perez's opener, then doubled his side's lead soon after the hour mark, but a Victor Osimhen brace in the final 20 minutes saw the visitors draw level and forced Brendan Rodgers's side to settle for a point.
Speaking to reporters after the game, the winger said: "They are a very good side and we knew it was going to be a tough game. But after playing some really good football, we will be disappointed. You don't want to be chasing points. You want to get them early. There's a little bit of a bitter taste not to come away with the win."
In the next round of group fixtures, the Foxes travel to Poland to take on Legia Warsaw, who defeated Spartak Moskow on Thursday to immediately take top spot in Group C.
CSKA Moscow 
Rubin 
Slovan Bratislava 
AC Milan 
Sparta Praha 
Atletico Madrid 
Valencia 
Barcelona 
Stade Brestois 29 
Bayer Leverkusen 
Red Bull Salzburg 
Bayern Munich 
Paris Saint Germain 
Inter 
RB Leipzig 
Manchester City 
Feyenoord 
Sporting CP 
Arsenal 
FC Rostov 
Spartak Moscow 
Aston Villa 
Juventus 
Celtic 
Club Brugge 
Dinamo Zagreb 
Borussia Dortmund 
Liverpool 
Real Madrid 
Monaco 
Benfica 
PSV 
Shakhtar Donetsk 
Lokomotiv Moscow 
Dinamo Moscow 
Anderlecht 
FC Porto 
AZ Alkmaar 
Galatasaray 
Dynamo Kyiv 
Viktoria Plzen 
Lazio 
Ludogorets 
Qarabag 
Lyon 
FC Heidenheim 
Chelsea 
FC Midtjylland 
Eintracht Frankfurt 
Manchester Utd 
Bodo/Glimt 
Nice 
Rangers 
Real Sociedad 
Ajax 
Tottenham 
Roma 
Fiorentina 
Pafos FC