You can use either one. "He believes everything being told to him" is used when you are referring to the things currently being told to him -- when there are specific instances of people now telling him things. It doesn't suggest that he always believes everything, just the things now being told to him.
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What about the differencesIf you say that you were being told, I get a picture in my mind of someone talking to you, and talking to you, and talking to you. I picture this as an activity that continued over some time.ofbetween I was being told and I was told?