The symbol * marks an ungrammatical sentence:
1. *The first thing I do is turning my TV on.
2. *The first thing I do is to turn my TV on.
3. The first thing I do is I turn my TV on.
4. The first thing I do is turn my TV on.
Copular constructions are like Math equations. The left side equals the right side: X = Y.
Sentence 3. is grammatical because "I do" (X) and "I turn" (Y) share the same structure.
3. I do = I turn
Both the left side and right side have a subject (I) and both the left and right side have an unmarked infinitive verb (do, turn). The same holds true for sentence 4.
4. I do = I turn
In contrast, sentence 1. and 2. are ungrammatical. The left side and the right side are different.
1. do * turning
2. do * to turn
"do" is an unmarked infinitive, whereas "turning" has -ing and "to turn" has the infinitive marker 'to'. Both -ing and to carry meaning, or add meaning to the base form of the verb, which makes them different from an unmarked infinitive, and the reason 1. and 2. are ungrammatical.