The heat victory was snared by Oscar Chelimo of Uganda in a time of 13m 24.24s.
On Thursday, Whanganui-raised runner Brad Mathas gave qualification a good crack in heat four of the men's 800m, but ultimately finished seventh in 1m 47.70s and was eliminated from the competition.
Mathas, who now resides in Melbourne, attacked the race with great intent, hitting the front on the first lap and taking the field through the bell in 54.46s.
The eight-time New Zealand champion then kicked from the front, opening up a small but significant gap entering the back straight, to raise hopes of a potential qualification spot.
However, Mathas started to fade as the field rounded the final bend and despite battling to the finish he exited the competition in seventh in a heat won by Djamel Sedjadi of Algeria in 1m 46.39s.
Running in the 5000m with Beamish but in heat two, fellow New Zealander Hamish Carson also missed out on the final.
Carson always faced a formidable challenge in a stacked heat, which included stellar talent in Norway's Olympic 1500m champion Jakob Ingebrigtsen and double World Indoor 3000m champion Yomif Kejelcha of Ethiopia.
Carson made a move from midfield up into the top six just before 2km to give himself every chance of qualifying for one of the five automatic spots.
However, following a pace injection from first Luis Grijalva of Guatemala then Kenyan Daniel Ebenyo, the field quickly split, and Carson lost contact with the lead group.
In boiling hot conditions, the 33-year-old, who hails from the Kāpiti Coast, plugged on gamely to cross the line 12th in 13m 37.62s.
Beamish finished 23rd overall out of the 41 starters, while Carson was 25th.