Ecc 3:19 For that which befalleth the sons of men befalleth beasts; even one thing befalleth them: as the one dieth, so dieth the other; yea, they have all one breath; so that a man hath no preeminence above a beast: for all is vanity.
Ecc 3:20 All go unto one place; all are of the dust, and all turn to dust again.
Ecc 3:21 Who knoweth the spirit of man that goeth upward, and the spirit of the beast that goeth downward to the earth?
These passages teach that the spirit of the animals go downward into the earth, and that the spirit of man goes upward. Animals therefore don't go to heaven. However I have heard it suggested that since death shall one day be destroyed that perhaps the animals themselves shall rise up in the resurrection at the last day. So then as Christians maybe we can see our dead pets again one day who knows.