If the word "the" isn't part of the name of the show, it's an easy decision - keep it lower-case.
If the word "the" is part of the name of the show, it's more difficult. This is a question of style, about which conventions vary and different people have different preferences.
In the UK, there is a newspaper called The Times. The word "The" is part of its name. When the Guardian and the Economist refer an article in the Times, they write:
an article in the Times
When the Times and the Telegraph refer to an article in the Times, they write:
an article in The Times
...although the Times writes The Times in italics, and the Telegraph doesn't. The Economist writes the word Times in italics (but not the definite article). The Guardian doesn't italicise either of them.