Expiring transactions run into the problem of opening up an attack vector against the chain because someone can submit a transaction (which results in it being propagated around the network, which is not free) and then they end up not having to pay anything in the end. In general, we want every transaction that is propagated to result in a non-trivial cost to the sender.
One option to work around this is to have an expired transaction still be includable, but it would just cost 21,000 gas and otherwise do nothing. The 21,000 gas is to cover signature validation, cold account loading, balance read, and balance write. Maybe it could be a little less, since there is guaranteed to be no second account loaded (recipient).