I had this issue with the last place I lived in. First beer I brewed there I used half spring water but had to use half tap water because I didn't have enough bottled water. It was a pale ale and it sort of had that plastic off flavor to it that worsened over time. I didn't think it was the water, I thought it was more a temp control issue, so I brewed a Heady clone with all tap water as my next beer. I looked after the fermentation temps clearly but this time the beer had an even stronger plastic smell and taste to it. So much that it was undrinkable and I had to dump the batch. One of the only ones I've had to dump in over 4 years of brewing.
The next batch I brewed with all spring water and the issue was totally gone. It was definitely the chlorine/chloramine in the tap water.
Campden tablets will help with chlorine but I'm not sure if it helps with choramine. From what I remember, and this was a while ago that I was dealing with this issue, choramine can only really be totally removed by pretty lengthy contact time with a carbon filter. Perhaps maybe that was the only way to get rid of choramine using only a carbon filter... I can't remember... in any case I'm sure you can find more info on here about filtering chlorine/chloramine out of your tap water.
I just always buy good water to brew with instead of messing with my tap water. Good water in, good beer out!
Camp Brew Co.
Primary: NHC APA, IOBA
On Tap: AmberHop ale-session version
Bottled: Bourbon Oak Brolic Strong ale