What would happen if an amoeba ran out of proteins?

I'm not so sure that it would die, rather than cease being a living entity. That sounds silly, but there's a difference between the two. Think about viruses in comparison to cells. Viruses contain all sorts of materials for life, yet we don't consider them living.

If an amoeba didn't have proteins and didn't make proteins, it really wouldn't be carrying out any of the functions of something that is considered living. There would be no proteins to replicate and organize the DNA. The plasma membrane would be mostly useless, as it would have no channels or pumps for various ions and molecules, both necessary and unnecessary. There would be no functional ribosomes and tRNA, and none of the organelles would be able to carry out their processes, as almost everything requires enzymes. You would essentially just have some DNA and RNA floating around with some sugars within some lipid membranes.

The cell wouldn't even undergo apoptosis, because that requires proteins.

So really, it would just be a bunch of the components of life in one place. It wouldn't be dead or alive. It would just be nonliving material.

Autolysis. It will produce proteolytic enzymes that will self-digest itself. What a way to go.