Person A mines 1 BTC. Since a Bitcoin has no intrinsic value (as distinguished from market value), Person A has not created any value. They have created $0 worth of goods.

Person A sells their coin to Person B for $1,000. A is $1,000 richer and B is $1,000 poorer.

B sells the bitcoin to C for $2,000. A is still $1k richer, B is $1k richer, and C is $2k in the hole.

C sells to D for $3k. A, B, and C are all $1k richer, but D is out $3k.

At no point in the chain is any actual value created or destroyed; it merely changes hands. So even though the price keeps going up, the total value across all participants never goes up. The sum of the gains and losses of all participants is $0.

(Note that I'm using "$" to represent a generic indicator of value. Feel free to substitute € or ¥ or gold or shares of Facebook stock or whatever.)

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There are two simplifying assumptions above, both of which make Bitcoin not exactly zero sum.

Good news first. Bitcoin can and does create some value in the same way that other currencies do: by facilitating mutually beneficial exchanges of goods and services. Commercial use appears to only be a tiny fraction of all Bitcoin transactions (and an even smaller fraction that wouldn't have been facilitated anyway using some other means of exchange), but it does happen so credit where credit's due.

The bad news, though, is that all of the above requires electricity. A lot of electricity, as it turns out. Both the initial mining and every transaction thereafter require someone, somewhere to burn some amount of power in order to generate proof of work. Since proof of work is explicitly designed to be useless, all that spent power is essentially deadweight loss.

I haven't seen any good numbers on the first recently (and I would love to have my assumptions proven wrong), but given the ginormous estimates of electricity consumed by the second I have a hard time believing the network as a whole isn't substantially worse than zero sum.