I have found a bug in my application code where I have started a transaction, but never commit or do a rollback. The connection is used periodically, just reading some data every 10s or so. In the pg_stat_activity table, its state is reported as "idle in transaction", and its backend_start time is over a week ago.
What is the impact on the database of this? Does it cause additional CPU and RAM usage? Will it impact other connections? How long can it persist in this state?
I'm using postgresql 9.1 and 9.4.
Since you only SELECT, the impact is limited. It is more severe for any write operations, where the changes are not visible to any other transaction until committed - and lost if never committed.
It does cost some RAM and permanently occupies one of your allowed connections (which may or may not matter).
One of the more severe consequences of very long running transactions: It blocks VACUUM from doing it's job, since there is still an old transaction that can see old rows. The system will start bloating.
In particular, SELECT acquires an ACCESS SHARE lock (the least blocking of all) on all referenced tables. This does not interfere with other DML commands like INSERT, UPDATE or DELETE, but it will block DDL commands as well as TRUNCATE or VACUUM (including autovacuum jobs). See "Table-level Locks" in the manual.
It can also interfere with various replication solutions and lead to transaction ID wraparound in the long run if it stays open long enough / you burn enough XIDs fast enough. More about that in the manual on "Routine Vacuuming".
Blocking effects can mushroom if other transactions are blocked from committing and those have acquired locks of their own. Etc.
You can keep transactions open (almost) indefinitely - until the connection is closed (which also happens when the server is restarted, obviously.)
But never leave transactions open longer than needed.