Student or Learner
- (On) what day were you born?
- (On) which day are you going?
- (On) what date did you start working?
- (On) which date are you starting to work?
I would like to ask if I should omit "on" or not.
Since other, more high-traffic responders have not replied, I'll throw in my two-cents worth.
1. (On) what day were you born? would be understood to ask on which day of the week you were born. "I was born on a Tuesday." In common, spoken American English, the question would more probably be, "What day of the week were you born on?" Or, far more commonly, "When is your birthday?" (Who cares what day of the week it was?) Here I think you do need the 'on'.
2. (On) which day are you going? means there is already an understanding that there is some limitation on the possible days of attendance, and the asker wants to know which of those days the listener will be there. The 'on' is not needed.
3. (On) what date did you start working?
4. (On) which date are you starting to work?
The only difference between #3 and #4 is that in number 4 we have limited the possible choices. As given, the 'on' should be included in each. Neither would be common in spoken American English.