Even now, GPUs are not used for the EVM. Mining is simply deliberate busywork, a way to spend computational resources with an easy way to prove that it happened in retrospect. Smart contracts and the EVM are totally done by the CPUs of every full node.
Post-Casper, GPUs likely won't be used at all. Casper does not require heavy computation, let alone deliberately wasteful ones. In a pure PoS world, validators (the PoS equivalent of miners) run smart contracts and the EVM on their own CPUs. In addition to transaction fees, they might be paid a subsidy of new ETH. (As far as I know, this has not been finalized.)
However, the current plan (as of May 2017) is for a stage of a PoS/PoW hybrid chain, where both Casper and PoW miners will coexist