why is there tungsten in the base metal? is it there intentionally as a hardsurface or put there accidentally during welding with the TIG process? Tungsten is extremely hard and is often used as a hardsurface application.Tungsten will not melt during most welding processes and will leave hardspots in your weldzone if it mixes with your filler metal. It should not cause cracking if welded with 309L unless there is ALOT of mechanical stress applied to the weld.As far as affecting the welding conditions, you may notice it in the weld puddle depending upon the concentration of tungsten in the parent material, but should still be weldable. However, if the tungsten is present for a hardsurface application, the deposited 309L weld metal will not have the same abrasion resistance.

How are you determining that there are tungsten inclusions in the (you stated base material, but did you mean...) weld material? Most methods of NDT used to determine this have accept/reject criteria to use to determine whether or not it must be removed.
John Wright