Upvote:2
Depends on what exactly you mean by "time".
Time spent in the air: that is indeed a function of direction, primarily because of the prevailing winds. For example BOS->SFO is about half an hour longer than SFO->BOS. The prevailing winds are indeed partially caused by the earth's rotation.
Difference in local time: You gain or lose time by crossing time zones. In the example above, you gain three hours from BOS to SFO and you lose them again on the way back. BOS-SFO takes 6 hours in the air but it's only 3 hours in local time difference. Flying from Shanghai to San Francisco you arrive actually 3 hours before your departure. A 12-hour flight and 15-hour time zone difference.
Relativistic time shifts due to speed differences: that theoretically exists but is way way way too small for any meaningful impact. See link in the comments.