I did note I was {thinking out loud}. :) Simply put, here is what you're asking for the plugin to do:
Step 1. Read the $lyrics.
Step 2. Print the first line from the $lyrics.
Step 3. Remember that you printed the first line, so you can print the second line the next time you run.
Step 4. Increment line count, rinse, lather and repeat.
There is nothing in a PHP script (which is all a plugin file really is) on its own which lets it get past Step 3. What's needed is some way to retain the information about the last line displayed. One way to do that in WordPress is to add a custom record to the options table (i.e. in the database). That's what my foreign text above is recommending.
angsuman, good point. Or if we make it a 2.0-only mod, we can stick it in usermeta. :)