It depends on how your water company treats your water. If it is using soda ash strictly for purposes of pH adjustment then you may have an excess of sodium in your water. Sodium in small amounts isn't harmful to plants, in fact they actually use miniscule amounts of it. If soda ash is being used for the purposes of reducing hardness then the Ca and Mg ions are basically being exchanged for Na ions. Some companies then remove the excess Na from the water. Some don't. In either case there is a good chance that your water may have too much Na for your plants. If you aren't experiencing any problems then it's probably safe to assume that your water company is removing most of the Na after treatment with soda ash.
This is an interesting question and unfortunately different municipalities use different methods based on the water chemistry being treated. This makes it very difficult to predict what problems a person may run into when using tapwater for their plants.