I came to that point last summer. Both health and financial circumstances made it very hard to keep the pets.
I began giving away the pets that were the easiest for me to be without (emotionally). The aquarium fish had to go. The turtoise. The little Siberian hamsters. - I kept those mostly because my small pupils like to get a visit of them sometimes.
The chicken and pigeons had to go next. It cost too much to keep them with too little in return. Both healthwise and financially.
Next to go are the pet birds. They are not expensive to feed in general terms, but for me, at the moment, any small saving will do. Still, I like to hear them twitter in the morning, so I have not given them away yet. I'll see in another month or two.
I am going to keep my cats, they are the pets I like the most, so it would be hard to be without them. They also keep rodents and snakes at bay.
If there is a war, and we cannot go to work for a long time, the cats will probably have to go too, but I hope that things will not come to that! The pet shop owner in the village is fully willing to take them any day - they are beautiful Persians...
I have never fed them the most expensive foods, but keep to medium-prized dry food of well-known trade marks. I did try cheap supermarket cat food, but two of my cats didn't feel well from it, so I am back to the well-known sorts again. It's worth it in terms of saving on vet bills.