How to use SSH to run a local shell script on a remote machine?

If Machine A is a Windows box, you can use Plink (part of PuTTY) with the -m parameter, and it will execute the local script on the remote server.

plink root@MachineB -m local_script.sh

If Machine A is a Unix-based system, you can use:

ssh root@MachineB 'bash -s' < local_script.sh

You shouldn't have to copy the script to the remote server to run it.

This is an old question, and Jason's answer works fine, but I would like to add this:

ssh user@host <<'ENDSSH'
#commands to run on remote host
ENDSSH

This can also be used with su and commands which require user input. (note the ' escaped heredoc)

Edit: Since this answer keeps getting bits of traffic, i would add even more info to this wonderful use of heredoc:

You can nest commands with this syntax, and thats the only way nesting seems to work (in a sane way)

ssh user@host <<'ENDSSH'
#commands to run on remote host
ssh user@host2 <<'END2'
# Another bunch of commands on another host
wall <<'ENDWALL'
Error: Out of cheese
ENDWALL
ftp ftp.secureftp-test.com <<'ENDFTP'
test
test
ls
ENDFTP
END2
ENDSSH

You can actually have a conversation with some services like telnet, ftp, etc. But remember that heredoc just sends the stdin as text, it doesn't wait for response between lines

Edit: I just found out that you can indent the insides with tabs if you use <<-END !

ssh user@host <<-'ENDSSH'
    #commands to run on remote host
    ssh user@host2 <<-'END2'
        # Another bunch of commands on another host
        wall <<-'ENDWALL'
            Error: Out of cheese
        ENDWALL
        ftp ftp.secureftp-test.com <<-'ENDFTP'
            test
            test
            ls
        ENDFTP
    END2
ENDSSH

(I think this should work)

Also see http://tldp.org/LDP/abs/html/here-docs.html