I googled and found a site on pet birds that said that they wouldn't use pomegranate wood with their birds. They didn't seem to have a definitive answer on pomegranate in particular, but they said it is very high in tannins.
On wood in general: I've been hanging out on a couple of German-language piggie forums, and I've gathered that in Germany, it is considered beneficial to offer piggies twigs regularly.
I've been using this site to check on toxicity of woods. It seems to be based on toxicity to humans, but it's a good start. (I'd use the German lists, but they hardly apply in the US!) Unfortunately, most of the woods around here are listed as toxic, so I'm trying to learn to identify the trees in our nearby greenbelt so I can bring the piggies home some twigs.