Python program to translate sentences from "bird language"
Considering that you have four concrete test cases, it would be a good idea to put them in a docstring as doctests.

Your iteration is awkward.  First of all, you never use i, so there is no need to call enumerate().  A better way to conditionally skip ahead is to call next() on an iterator over the phrase.

def translate(phrase):
    """
    Translate the phrase from bird language.

    &gt;&gt;&gt; translate('hieeelalaooo')
    'hello'
    &gt;&gt;&gt; translate('hoooowe yyyooouuu duoooiiine')
    'how you doin'
    &gt;&gt;&gt; translate('aaa bo cy da eee fe')
    'a b c d e f'
    &gt;&gt;&gt; translate('sooooso aaaaaaaaa')
    'sos aaa'
    """
    phrase = iter(phrase)
    result = ""
    for letter in phrase:
        result += letter
        if letter in VOWELS:   # Vowel. Skip the next two characters.
            next(phrase)
            next(phrase)
        elif letter.isalpha(): # Consonant. Skip the next character.
            next(phrase)
    return result


A much more compact solution, though, would be to perform a regular expression substitution.

import re

def translate(phrase):
    return re.sub(r'([aeiouy])\1\1|([a-z])[aeiouy]', r'\1\2', phrase, flags=re.I)