The EIP86 makes it very easy to send transactions from an address without ETH. This allows miners to easily accept transactions that do not pay fees in ETH; opening up a world where users don't need to pay fees for their transactions. Does this obsolete ETH, primarily used for gas?
What exactly is happening
Ethereum improvement proposals (EIPs) are the standard method for defining new protocol specifications and features for the Ethereum network. Similar to the structure in Bitcoin, these proposals follow from draft, to accepted status as consensus forms on the proposal in the community. Accepted proposals are expected to be included into the next hardfork. The EIP86 that includes the features allowing for ETH free transactions has been accepted and should be included in the next Metropolis hardfork planned for the next month or two.
The EIP86 outlines the abstraction of transaction origin and signature. This change to the Ethereum protocol is part of the attempt to abstract account security from the standard ECDSA + nonce scheme used to currently secure accounts. This will allow for the use of more difficult ed25519 and Schnoor signatures, quantum-resistant signatures or whatever complex signature scheme users want. Ethereum is moving towards a future where all accounts are 'contracts' that can both pay for their own gas and define their own cryptography and security schemes. This makes Ethereum much more flexible as a platform and allows its consensus mechanism to support many more types of application on the public chain.
The abstraction of the transaction origin allows transactions to originate from the 2^160 - 1 'anyone can send' address. In simple terms it allows you to send a transaction from nowhere; normal transactions must originate from your account or contract with ETH for the transaction gas (fees). By allowing transactions to originate from this 'entry-point' address, there is no ETH available for fees - by design. This allows miners to choose to include these transactions for reasons other than the associated gas. These transactions could be set up to pay miners in any other token or asset. The fees for these types of transactions can come from a number of different sources - not necessarily the transaction sender.
The owner of a contract could incentivize miners to mine transactions pertaining to the contract using their own means. Essentially allowing for application developers to pay for their users. This can enable a EOS like model to develop on Ethereum.
This gives miners more control over what transactions they wish to mine and for what cost. I see this feature allowing for very unique applications running on Ethereum with their own incentive structures for including transactions into blocks. If a miner is concerned about the free to use nature of a social network like steem, and that miner is compensated using built-in asset backed by economic game theory of the social network, the users won't need to pay for their vote transactions.
Does this mean my ETH is going to be less in demand?
A worry expressed by some is this EIP will cause ETH to be less in demand and therefore less valuable if it is not used to pay the gas fees for the network; nothing is obligating the use of ETH. This is a false worry. Ether will be the only token that can be used for staking on Casper POS when launched.
It does not make any sense to have a currency agnostic staking system. The type of consensus system that would be needed to support assets with independent fluctuations in their value is too complex and vulnerable to failure. The stake value of Ether will be determined by the value of all other transactions on the network, regardless of if they are in Ether or not.
Interestingly, most of this 0-ETH transaction utility is possible today. You can do most of above with a self-mined transaction or by doing an off-chain deal with a miner. Miners are free to include transactions of any kind they wish - even those without fees. EIP86 simply makes this easier and allows other miners to easily accept different types of transactions based on their client-side rules. Not everyone is able to mine their own transactions, however.
While miners are going to be free to accept whatever they choose in the future, the universality of ETH will likely remain a preferred method. On the ETH chain, all other tokens are essentially pegged to ETH values so I certainly expect many token markets to flourish against ETH. More transactions on the network, even if not directly using Ether will benefit the general network effect of Ethereum. ETH will always have utility over other tokens - even if that value is only staking.