I don't know what kind of paper Thangles uses... but when I paper-piece, I use vellum. Ironing it seems to make it brittle and tear off easily.
Having said that, I'm like an earlier poster - most of the time I don't sew through the paper, I fold on the line and sew right next to it. The only time I sew through the paper is when I think the block/fabric needs stabilizing.