Is there a secret tunnel that connects the Thirumalai Nayakar Mahal and Meenakshi Amma temple in Madurai?

  Lakshmi,

May not be.

1. Meenakshi Temple in present form is reconstructed in 14th century by Vijayanagara Emperors. Madurai Naickers become feudatory kings of Vijayanagara from 1529.
2. When Tirumalai Naicker (1623–1659) constructed his Mahal, the Capital was Tiruchirappalli. Madurai became capital in 1634, so as construction of Tirumalai Naicker Mahal.
3. Political events lead to the change in Capital from Trichy to Madurai in South are: on hearing the fall of Vijayanagara Empire, Ramnad revolted, Dutch and Portuguese threatened Tenkasi Pandiyans who were feaudatory to Vijayanagara, Dutch occupied Tiruchendur Temple, looted the treasures and idol, and set fire the temple. This lead to the expedition to South and change of capital.
4. Tirumalai Naicker made additions in Madurai Meenakshi Temple like 1000Pillar Hall in the temple etc and Madurai Theppa Kulam. He started Meenakshi Kalyanam and Azhagar festival to maintain social harmony among Shaivites and Vaishnavites.
5. Madurai Naickers constructed Forts in Dindigul and Trichy, which are their strategic bases, not Madurai.
6. Rani Mangammal (1689–1704) who is considered as the Constructor of Roads and Mandapams for resting(Mangammal Salai) shifted back the capital from Madurai to Trichy in 1695 and used Present day Gandhi Museum as her Palace in Madurai.

So it is unlikely, any secret tunnel connecting Madurai Meenakshi Temple and Tirumalai Naicker Mahal.