Sheep often lie down, Tegan, to digest their food. Sometimes they don't bother standing up again for the next course, but just eat what is within reach from their comfortable lying-down position.
I don't think their owner would use a lethal dye on them. Sheep are too valuable to kill off for a short joke while the Tour is on. There are various coloured dyes which are mixed with sheepdip (done round about now to discourage fly strike and other nasties) so that the dipped ones don't get two dips while others miss out.
Shearing is done about now too, at least in the Scottish Borders it is. Before the sheepdip so that a) the fleece stays white ish and b) a shorter coat does not absorb and use up so much dip.