Kicking off with what should be the most obvious point against Toy Story 4 - it's painfully and utterly unnecessary. Toy Story 3 was the ending of the series. With Andy passing on his toys to pre-schooler Bonnie the story of Woody and Buzz was wrapped up in a thematically tight manner. That was a central element of the film's success; it served as a cap on the childhood of those who grew up with the earlier movie and a reminder of how life doesn't end when you grow up to everyone else. To revisit them a few years down the line will utterly discredit this development. Of course the fact that the gang are with a new owner means that narratively there's potential to extend their adventures, but that doesn't mean there's justification. The Lord Of The Rings ends with most of the Fellowship alive and in Middle-Earth, but there's no desire to continue that; the story is over. Even if Toy Story 4 ends up being a worthy entry in the series (and we'll see in subsequent points that's unlikely), it'll still ruin the resolute ending Toy Story 3 gave everyone.
