While Pant and Amre were fined their entire match-fee, Thakur was penalised 50 per cent of his match-fee, IPL said in a release.
The 10 teams in the league, following the addition of Gujarat Titans and Lucknow Super Giants, will be bidding heavily for the Indian players in a list of 590 cricketers, including 227 overseas candidates up for grabs during the two-day event.
Thakur sent back Temba Bavuma (51) to record his maiden five-for in the longer version.
The seam bowling all-rounder, who turned 30 on Saturday, had played a big role in CSK's triumph with figures of 3/38 in the final.
The 29-year-old Thakur has impressed with his 18 wickets in the ongoing IPL for Chennai Super Kings.
Pandya is a key player for India and it will upset the balance of the team if he's unable to bowl.
Thakur's contribution -- two fifties and three crucial wickets -- was more spread out and with both the ball and the bat.
Virat Kohli's men, who were shot out for 191 on the first day and conceded a 99-run first innings lead, displayed fighting spirit and resilience to stage an amazing turnaround.
Shardul Thakur smashed the fastest Test fifty on English soil.
Thakur, who picked up four wickets in the first Test at Trent Bridge, did not train with the team this week but should be fit for the third Test at Headingley.