But I don't think tigers are shut in the cage is good for them.

如果按你的写法应该是

But I don't think shutting tigers in the cage is good for them。才对

因为你的句子里把tigers作为宾语从句里的主语，首先句义不符合逻辑，其次那样会使从句里的主语tigers和谓语is单复数不一致。