That is, can you build a Turing Machine that, if given a Turing Machine as input, can decide (or at least recognize) if the inputted Turing Machine has an infinite number of strings in its language?
Similarly, can you recognize or decide if a Turing Machine has an uncountably-infinite language?
To elaborate somewhat on Yuval's comment:
(1) Some languages have an infinite number of strings, some do not. This makes the property non-trivial, and Rice's theorem says that recognizing whether the language of a Turing Machine has a non-trivial property is undecidable. This problem is undecidable.
(2) All languages are countable; none are uncountable. This makes the property trivial. Deciding whether the language of a TM has a trivial property is always decidable: if it's a property all languages have, then answer yes; if it's a property no languages have, answer no.
(A word of general caution, though it doesn't apply here. Rice's theorem applies only to properties of languages themselves, not properties of the TMs that might accept them. This is an important distinction. Rice's theorem only works for properties that have to do with the language and strings it contains.)
For reference: Rice's Theorem
