If all animals were to die, would all plants die too?
Animals are not the only organisms to respire without photosynthesizing. Fungi, many other eukaryotes and many bacteria respire too (and do not photosynthesize).

Now, if all animals were to suddenly die, there would sure cause a drastic environmental change for plants. Whether or not that would lead all plants to extinction is untestable and it is therefore impossible to give a definite answer. This type of question would be better suited for worldbuilding.SE. I personally doubt all plants would disappear.