Yes of course, but in this case the coins give value to the blockchain itself. If we remove the economic part, do you really think someone will use that system?
Not very likely with a PoW implementation as the economics of the "coin" is what enables the "work" to get done (unless of course the mining is all government provided).
But with other low-cost consensus systems (and so far none have really being proven to be anywhere near as reliable as Bitcoin) the motivation could be the value of what else the system can provide (e.g. a decentralised "blogging" implementation) so assuming the cost to participate is negligible then the utility of the software platform could provide the reason for people to support the network (much like the motivation for running a ToR node).
