First of all, how would you get to NY if you dropped the first leg of the trip? was it cheaper to book the SFO-JFK--LGA-DEN-YVR vs booking the LGA-DEN-YVR as one?
If you are removing the first leg of a ticket, you are almost traveling or departing from a whole new city making a new fare.
The only time I have had segments removed from my ticket was on a MR/DO trip to SFO where i was flying IAD-SFO-LAS-SFO-IAD. When I booked the ticket i only wanted to fly IAD-SFO-IAD but that cost $500 more. I was able to get the SFO-LAS-SFO portion of the ticket removed in the RCC because the SFO-LAS flight was delayed enough to make me miss my fly back to SF for the dinner. So the RCC CSR called a SD and was able to remove the segment with out a problem. I just stayed in SF and went to the Giants game before the DO.