My Mod Podge coat is still sticky. What do I do?
Mod Podge takes a very long time to cure; around four weeks, depending on humidity and coat thickness.

To properly use Mod Podge, you need to apply multiple thin layers (not thick ones), allowing them to dry completely between each coat. According to Meaningful Mama,


  If you don’t let [the Mod Podge] dry completely between coats, your project can end up more sticky and may bubble a bit.


"Dry" is not "cured," however; while it will only take an hour or so (again, depending on humidity) for each coat to dry, you need to leave the pieces alone for roughly a month to become fully cured and hardened. Once fully cured, the Mod Podge will be a durable coating resistant to damage, but during the cure time, it will remain soft and susceptible to damage, including shifting and peeling.