I'd say that for anyone who can sew, polarfleece squares (~ 2 feet by 2 feet or so) are great cage liners. Or, polarfleece on one side and fabric on the other side. You can do a quilting type stitch to stick them together and they can be laundered over and over. The hospital where I used to work had a very generous lady who used to make the polarfleece cage liners with different (seasonal) fabrics on the other side and they were the perfect size for lining cages, and they typically held up fairly well to laundering in our commercial machines.
We don't have those where I work now (and I don't sew or have a machine). We mostly use towels and some blankets, and we take all sorts of donated towels and blankets and some sheets that people don't want anymore. Crocheted or knitted things, in general, do not hold up that well. Remember, these are being washed DAILY if not more, and getting all manner of puke, poop, pee, etc. on them. Claws and paws can get into yarn and rip them, and ANY loose threads or strings are a hazard since a pet could swallow them, so once anything knitted has any holes or is unraveling, even in one tiny spot, it would have to be trashed.
I think it's really admirable to want to help a shelter and I am sure they ARE grateful even if you're providing something knitted that may not last as long as other types of bedding. It would be incredibly rude (and I can't imagine them saying) that they didn't want your handknitted things. A gift is a gift! However, I don't think that knitted items are very practical for use as cage bedding when compared to the amount of effort required to make them. Just my 2 cents...
If you're donating to a specific place, you might want to ask them what they could use the most. If they say that your knitted or crocheted things work well for them, then go for it!