Examples of biological turing machines
There have been several Turing machines constructed using DNA computing. One of these machines has been used to solve the boolean satisfiability problem, another was used to solve the bounded post correspondence problem, both NP-hard combinatorial problems which are difficult for conventional computers to solve.
Also, a DNA computer was constructed that plays the optimal strategy in tic-tac-toe.

DNA computers store information in DNA sequences and perform calculations by DNA baseparing and other molecular biology strategies. In principle, DNA stores Information much denser than any modern hard drive and uses orders of magnitude less energy per calculation than a conventional computer ($10^{-22}$ vs $10^{-10} \frac{J}{operation}$) however, so far i/o is slow and calculations are comparatively error prone, limiting the practical use of DNA computing.

Also, a recent article has demonstrated that molecular motors can perform basic addition and solve the NP-complete subset sum problem by exploring nanolithographically fabricated networks. This could also, in principle, be used to construct a turing machine.