Most likely you have a 'negative' pressure condition in the mechanical room or even the whole house. This is a case where adequate combustion/make-up air does not exist or is insufficient to provide proper ventilation of the water heater. As a result, air migrates down the chimney in an attempt to equalize the pressure differential of outside vs inside.
Some things that can exacerbate this condition are: using a whole-house fan, window fans blowing to the outside, using exhaust fans, using a gas or electric clothes dryer.
As even older homes are now becoming 'tighter' to outside air infiltration through remodelling, window and door replacements, and insulating, one thing that is often overlooked is combustion/make-up air.
If you do have adequate combustion air (minimum 4" for only the WH), you may want to look at a few things with the venting and the chimney/liner itself.
First off, connector rise from the draft hood of the WH to the elbow above it. This should be as high as possible while still maintaining 1/4" per foot slope up to the chimney. This is how the flue gasses gain upward momentum to 'temper' the chimney. 'Temperring' of a chimney is heating it up to sufficiently draw the flue gasses upward and out the top by gravity (heat rises).
Second, horizontal flue distance. This is generally limited to 1/3 of the chimney's height. Addtionally, horizontal offsets are limited to 135° total, if not less in some localles.
Third, chimney height above roof. A chimney that is too low vs the roofline will allow winds to interfere with proper venting. A general rule of thumb is that from the top of the liner cap, there shall be not less than 4 feet horizontally to the roofline.
Now, where I'm from, we're not allowed to put in 4" liners, 5" is the minimum. Additionally, there is much discussion as to whether or not an atmospherically vented water heater (sub 200° flue gas after the draft hood) can sufficiently temper a lined chimney on its own.
My opinion is that it can't in our climate. The reason is because of the draft hood of the heater. Excess air is mixed in at the bottom of the draft hood which lowers the flue temperature, this is done as a safety measure to separate the burner in the WH from the flue. Otherwise every time the wind blew your pilot would go out, and if the flue were to be blocked, adequate air can still move within the WH to mitigate CO concerns, the burnt gas will just spill out of the draft hood. When you couple having to maintain these safeties with the larger chance of spillage, a power-vented water heater is by far safer.
Power-vent water heaters should go hand in hand with 90+ furnace changeouts. Some places mandate that, they all should.