Burnley midfielder Hendrick stars for Ireland in Denmark
Fellow Clarets Kevin Long and Robbie Brady also started the game in Aarhus.
Nicolai Jorgensen hit the post with a gilt-edged second-half chance after poor defending from Richard Keogh, with Yussuf Poulsen also missing a sitter.
Denmark were very wasteful with their 76% possession, and had just one shot on target from 26 attempts.
The Republic's relegation to League C in the new competition was already confirmed before Monday's game, in which their poor run of form extended to only one win in 11 games.
The Republic Of Ireland will now wait for next months Euro 2020 qualifying draw, where they will be placed in pot 3.
The Burnley trio will now return to the club ahead of Monday’s clash with Newcastle United at Turf Moor.