The solution is to use loose tea. However, I've realised that many teas come only in tea bags!
Moral Fibres has a summary of different brands of tea bags that do not contain plastic - Teapigs, Twinings pyramid teabags, Pukka (each teabag is individually packaged in a plastic sachet though), Tetley (only string & tag teabags used for catering), PG Tips pyramid teabags.
Abel & Cole are plastic free according to their own website.
The Independent has a few others to add to the list - Brew Tea Co, Hampsted Tea, We are Tea, Eteaket, Nemi, Roqberry, Good & Proper, T2.