Yes, since reads destroy the data, each read causes a write. The chip will handle 100 trillion read/writes. So yes, the chip will wear out rapidly if you do a lot of reads.
I can't really think of a use case where this could plausibly come up. It would take 3 years if you read the same memory location (uncached) 1 million times a second.
There are multiple factors that limit the number of writes that FRAMs can handle: changes in crystal structure as Ti ions replace O, mobile ions collecting at grain boundaries, and something to do with 90ยบ domains.