A picture of the vent pipe setup would be helpful. Assuming that you have an induced draft motor equipped furnace, both vent pipes (from furnace and WH) must be connected together with a WYE type connector – check this post Chimney Flue.
Otherwise, furnace induced draft motor might be forcing air / exhaust gases into the water heater vent pipe. Whenever the water heater burner starts operating first and heats up the chimney, the furnace exhaust gases have some extra draft help from already warm chimney and instead of flowing back into the WH vent pipe, they continue into the chimney.
If you have a "Y" type connector properly installed (I’ve seen them up side down), section of your chimney might be partially blocked, and before it picks up proper draft, it spills some of the exhaust gases back into the WH vent pipe.
It might be also an oversized, cold chimney that causes back-draft. One other thing that I can think of is a missing or collapsed WH draft hood… ?
Please, let me know.