It is possible to transfer the start dates from the Projects/Tasks in Web TimeSheet to Web Resource during Integration by having the option 'Overwrite Web Resource Project dates with Web TimeSheet project dates' selected as seen in the 'WR4.PNG' screenshot attached to the solution. This article in such a scenario describes how such a Task Start/End date is transferred in Web Resource.
The following scenario is explained in the below example:
The Project structure of the project to be transferred.
Sub T2 which has start/end dates.
The task Test whose Parent is Sub T2 which does not have Start/End dates.
Up on transferring the whole project from Web TimeSheet to Web Resource, one can see that the same Task 'Test' now in Web Resource has Start and end dates as "1st September 10" to "15th September 10" (Please refer to the screenshots) which is nothing but the Start/End dates of its parent task 'Sub T2'.
Therefore, whenever there is a Start/End date of a task missing during the Integration from Web TimeSheet to Web Resource, these dates are copied from its parent with the option 'Overwrite Web Resource Project dates with Web TimeSheet project dates' selected.