Is the phrase "the alive animal" grammatically correct?

The animal was alive is a correct sentence. The alive animal is a noun phrase and therefore not a sentence, since at the very least it is missing a verb.

Having said that, I interpret the question as asking whether it is possible to place the adjective alive in front of a noun. In other words, can alive be used attributively? Most adjectives can be used both attributively (before the noun) and predicatively (following a copula such as the verbs to be, to seem, to look). So happy can be used both attributively and predicatively:

• The happy child [played in the sand].
• The child looked happy.

But some adjectives can only be used predicatively; they cannot be used attributively. Alive is one such adjective, so the alive animal is ungrammatical. There are many other non-attributive adjectives beginning with the letter a. For example, it is ungrammatical to say:

• *the afraid child ..
• *the asleep child ..
• *the alone child ..

Other adjectives can only be used attributively; they cannot be used predicatively. For example:

• my maiden aunt (*my aunt is maiden)
• my sole pleasure (*my pleasure is sole).