Heena, who had finished 12th in her maiden Olympics in London 2012, failed to recover from a poor start and finished 14th in a field of 44 shooters.
In the men's trap qualifiers, Manavjit Singh Sandhu and Kynan Chenai finished 17th and 19th respectively after the first day.
Sidhu, the 26-year-old 2010 Guangzhou Asian Games silver medallist, shot an overall 380, which included 13x, to become the fourth Indian shooter to be eliminated on day two of the competition.
The 26-year-old Heena will now try to make amends when she competes in the 25m pistol event scheduled to be held on August 9.
In men's trap shooting qualification, Sandhu and Chenai could come up with scores of 68 and 67 respectively in the three rounds after a total of 33 shooters participated on day 1 with each athlete taking 75 shots each.
Sandhu scored 23, 23, 22 out of 75 while Chenai posted 22, 23, 22 in the three rounds at the Olympic shooting centre here. The duo will return tomorrow for the second day of qualification where there will be two more rounds of 25 targets each.
The cut for the eight team finalists was applied at 384 with seven closest to the centre circle, which was shot by Sonia Franquet of Spain.
In men's trap shooting, Sandhu was placed at 14th while Chenai lay 28th after the completion of the first round. The former improved a place with a score of 46 in the second round, while Chenai too rose to the 16th spot with a score of 45.
Yesterday, the duo of Apurvi Chandela and Ayonika Paul failed to qualify in the women's 10m air rifle event.
Apurvi a World Cup Final silver-medallist and the Commonwealth Games gold winner, finished 34th with a score of 411.6 while Ayonika ended at the 43rd place with a final score of 407.
It was Jitu Rai, who could make the final in the 10m air pistol, but finished a disappointing eighth.
