Right now Trans Mountain supplies the lower mainland with various fuel products. At Abbotsford there is a split which diverts some of the product coming down the pipe to refineries in Washington state.
After the twinning Trans Mountain will convert to 100% bitumen for export and BC's south coast will then have to resort solely on exports of fuel from the US for it's needs. 
So to answer your question, yes BC uses fuel and oil shipped by TMX but it won't once it's twinned.