Anhydrous Ammonia is utilized to melt the lithium strips in the meth process, correct?
No, that is wildly incorrect. It is used to dissolve the lithium, and due to certain properties of anhydrous ammonia, this dissolution creates solvated free electrons in the solution, and these are in turn the active reducing species in the Birch Reduction.
Can liquid nitrogen be used in place of it?
Not at all. Liquid nitrogen is a very bad solvent for most substances, due to its non-polarity. Also, it cannot harbor any solvated free electrons, the only other solvents besides ammonia which can do that are certain lower alkylamines (this has been discussed a lot in the Stimulants forum, try to search for "birch AND amines" and follow the threads which turn up.