Check out this Turing machine simulator: http://morphett.info/turing/turing.html
Read about recombinase-based circuits: http://www.nature.com/nbt/journal/v31/n5/full/nbt.2510.html -- could recombinases be used here to "write" to the DNA tape?
Kurt Whittemore's iGEM idea for a splicing-based Turing Mahcine: http://www.scribd.com/doc/8703987/Biological-Turing-Machine-Idea
A theoretical, polymer-based Turing Machine: http://rsfs.royalsocietypublishing.org/content/early/2012/03/21/rsfs.2011.0118.full