Murray's encounter with Frenchman Nicolas Mahut was halted with the world number two leading 6-3 2-2.
With Murray's match unable to finish due to the consistent rain Tsonga's clash with compatriot Edouard Roger-Vasselin did not start and the fourth seed now faces a packed schedule, with doubles matches also abandoned for the day.
Of the matches that did reach a conclusion there was success for second seed Tomas Berdych and fifth seed Marin Cilic, who recorded straight-sets victories over Thiemo De Bakker and Ivan Dodig respectively.
Seventh seed Alexandr Dolgopolov also progressed courtesy of a 6-3 4-6 6-4 win over Santiago Giraldo but it was not all plain sailing for the top players on the third day.
Four-time Queen's winner Lleyton Hewitt sent 10th seed Grigor Dimitrov packing with a comprehensive 6-4 6-3 win, while Dan Evans gave the British public something to cheer as he dispatched Finnish 13th seed Jarkko Nieminen 6-4 6-7 (6/8) 6-4.
Benjamin Becker set up a third-round tie with Dolgopolov following a 7-6 (7/2) 7-5 victory over 12th seed Lukas Rosol, while eighth seed Kevin Anderson pulled out of the tournament due to a shoulder injury and his replacement Rohan Bopanna lost 6-4 6-3 to Kenny de Schepper, who goes on to face Denis Kudla for a quarter-final berth.
Grega Zemlja will face Berdych in the third round following a 7-5 6-4 win over Sergiy Stakhovsky, with Marinko Matosevic waiting for either Murray or Mahut in the third round after Michael Llodra withdrew following a hamstring injury.
As well as Murray's match on centre court, the clash between qualifier Feliciano Lopez and 11th seed Julien Benneteau on court one and Denis Istomin's match against Igor Sijsling on court two were also interrupted by the rain and remain unfinished.