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Needed isn't a word I would use but yes, the 7-day JR Rail Pass would save you some money on this itinerary.

The trick here would be to activate the Pass for the day or two before you go to Kyoto. From there, you could use it for the trip to Kyoto, a round trip day to/from Nara, the trip from Kyoto to Osaka, the trip back to Tokyo and any other rail travel on JR lines during that 7-day period. That all adds up to a modest savings even though you won't be going back out to NRT for the flight home. I would get one just for the convenience even if the savings were only break-even.

Have you thought your itinerary through? Kyoto and Osaka are only minutes apart on the shinkansen (bullet train) and not much more on local trains. They are easily day-trippable from one to the other. We usually suggest staying in one or the other so as to avoid moving hotel more than necessary. Day trips from either one are easy with a Rail Pass to Kyoto/Osaka, Nara, Horyuji, Hikone/Lake Biwa, Himeji (best original castle in the country) and even a long day to Hiroshima.