Insert SQL statements via command line without reopening connection to remote database

Answer to your actual question

Yes. You can use a named pipe instead of creating a file. Consider the following demo.

Create a schema x in my database event for testing:

-- DROP SCHEMA x CASCADE;CREATE SCHEMA x;CREATE TABLE x.x (id int, a text);

Create a named pipe (fifo) from the shell like this:

postgres@db:~$ mkfifo --mode=0666 /tmp/myPipe

Either 1) call the SQL command COPY using a named pipe on the server:

postgres@db:~$ psql event -p5433 -c "COPY x.x FROM '/tmp/myPipe'"

This will acquire an exclusive lock on the table x.x in the database. The connection stays open until the fifo gets data. Be careful not to leave this open for too long! You can call this after you have filled the pipe to minimize blocking time. You can chose the sequence of events. The command executes as soon as two processes bind to the pipe. The first waits for the second.

Or 2) you can execute SQL from the pipe on the client:

postgres@db:~$ psql event -p5433 -f /tmp/myPipe

This is better suited for your case. Also, no table locks until SQL is executed in one piece.

Bash will appear blocked. It is waiting for input to the pipe. To do it all from one bash instance, you can send the waiting process to the background instead. Like this:

postgres@db:~$ psql event -p5433 -f /tmp/myPipe 2>&1 &

Either way, from the same bash or a different instance, you can fill the pipe now.
Demo with three rows for variant 1):

postgres@db:~$ echo '1  foo' >> /tmp/myPipe; echo '2    bar' >> /tmp/myPipe; echo '3    baz' >> /tmp/myPipe;

(Take care to use tabs as delimiters or instruct COPY to accept a different delimiter using WITH DELIMITER 'delimiter_character')
That will trigger the pending psql with the COPY command to execute and return:

COPY 3

Demo for for variant 2):

postgres@db:~$ (echo -n "INSERT INTO x.x VALUES (1,'foo')" >> /tmp/myPipe; echo -n ",(2,'bar')" >> /tmp/myPipe; echo ",(3,'baz')" >> /tmp/myPipe;)INSERT 0 3

Delete the named pipe after you are done:

postgres@db:~$ rm /tmp/myPipe

Check success:

event=# select * from x.x; id |         a----+-------------------  1 | foo  2 | bar  3 | baz

Useful links for the code above

Reading compressed files with postgres using named pipes
Introduction to Named Pipes
Best practice to run bash script in background

Advice you may or may not not need

For bulk INSERT you have better solutions than a separate INSERT per row. Use this syntax variant:

INSERT INTO mytable (col1, col2, col3) VALUES (1, 'foo', 'bar'),(2, 'goo', 'gar'),(3, 'hoo', 'har')...;

Write your statements to a file and do one mass INSERT like this:

psql -h remote_server -U username -d database -p 5432 -f my_insert_file.sql

(5432 or whatever port the db-cluster is listening on)
my_insert_file.sql can hold multiple SQL statements. In fact, it's common practise to restore / deploy whole databases like that. Consult the manual about the -f parameter, or in bash: man psql.

Or, if you can transfer the (compressed) file to the server, you can use COPY to insert the (decompressed) data even faster.

You can also do some or all of the processing inside PostgreSQL. For that you can COPY TO (or INSERT INTO) a temporary table and use plain SQL statements to prepare and finally INSERT / UPDATE your tables. I do that a lot. Be aware that temporary tables live and die with the session.

You could use a GUI like pgAdmin for comfortable handling. A session in an SQL Editor window remains open until you close the window. (Therefore, temporary tables live until you close the window.)

I know I'm late to the party, but why couldn't you combine all your INSERT statements into a single string, with a semicolon marking the end of each statement? (Warning! Pseudocode ahead...)

Instead of:

for each line  sql_statement="INSERT whatever YOU want"  echo $sql_statement | psql ...done

Use:

sql_statements=""for each line  sql_statement="INSERT whatever YOU want;"  sql_statements="$sql_statements $sql_statement"doneecho $sql_statements | psql ...

That way you don't have to create anything on your filesystem, do a bunch of redirection, run any tasks in the background, remember to delete anything on your filesystem afterwards, or even remind yourself what a named pipe is.