Raonic, the fourth seed in the tournament, suffered an injury-plagued 2019 which saw him miss the US Open and Canada's run to the Davis Cup final.
However, his new year is off to a losing start after falling to a 7-6 (4), 7-6 (4) defeat against French qualifier Corentin Moutet.
Tsonga, who won the Qatar Open in 2012, will not be joining them in the last eight after suffering a 7-6 (9), 6-1 defeat against Serbia's Miomir Kecmanovic.
Elsehwhere, World No. 23 Andrey Rublev found the going in Doha much easier, cruising to a 6-4, 6-2 victory in just 69 minutes against Mikhail Kukushkin, while there were also straight-set wins for Pierre-Hugues Herbert and Marton Fucsovics.