Do I need to overlay plywood on my drop-in tub deck before installing tile backer?
No, you don't. Doing so could be a way to stiffen and reinforce the framing, but it's not required. 

I'd be sure your corner laps are all secure and well supported. The left end in particular concerns me a little--that's a fairly long span for flat lumber. If there's any chance of movement, add framing underneath and screw things together well. Construction adhesive would remove all doubt. 

The key is to eliminate movement (flex) under the load of the tub users. (The tub itself shouldn't rest on the deck at all.) If you can see or feel movement under your own weight, it needs work.