Technically, it's not possible to run an Ethereum blockchain without Ether, since the two are tightly coupled at the protocol level. However, it is possible to develop, test and demonstrate Ethereum Dapps without using valuable Ether.
Most developers use blockchain emulator such as testRPC and/or private chains for development purposes, and then deploy on the public testnet for public testing and demonstration.
In all those scenarios, the Ether seems to exist and the accounting unfolds but the Ether is of no actual value.
By far the easiest procedure (at least easiest to describe IMO) is to use the testnet. This is just a matter of instructing the Mist browser and/or the geth/parity node to use testnet instead of mainnet. Mining on testnet is considerably easier than mainnet, which means one can usually earn an ample supply of Ether for experimentation with minimal effort and no expense.
Hope it helps.