In these simple examples, you want to have consistency in tense across the entire sentence. The easiest one is
Yesterday, I felt like I was a bird.
"Yesterday" establishes that this happened in the past. "felt" and "was" are both in the past tense and match the established tense.
Today, I felt like I was a bird.
Today, I feel like I am a bird.
are both valid, since "today" can refer to earlier in the day or the present. In the former sentence, using the past tense of both "to feel" and "to be" implies that it occurred earlier today and is not still ongoing. In the latter sentence, the feeling is still occurring, so you use the present tense of those verbs: "feel" and "am".
Take note of @Maulik V's comment, too. "I was"/"I am" aren't required for the sentences to be correct.