In Game Theory, Tragedy of the Commons is a scenario where a market fails because a common good is produced in lower quantities than the public desires, or consumed in greater quantities than desired.
A usual example is a pollution - it is in the public's best interest not to pollute, but every individual has the incentive to pollute (e.g., because burning fossil fuel is cheap, and individually each consumer believes he doesn't affect the environment much).
The relevance to Bitcoin and all Proof of Work altcoins is a possible market failure that could happen in the future when the block rewards from mining drop near zero.
In the current design of Bitcoin and Proof Of Work cryptocurrencies, the remaining reward for miners will be just the Transaction fees. Miners will accept transactions with any fees (because the marginal cost of including them is minimal) and users will pay lower and lower fees (in the order of satoshis).
It is possible that the honest miners will be under-incentivized, and that too few miners will mine, resulting in lower difficulty than what the public desires.
That prospect could result in various 51% attacks happening frequently, and the Bitcoin will not function correctly.
So to sum it up: The tragedy of the commons in cryptocurrency context is where miners are driven out of the network as profitability falls on decreasing transaction fees available.