I was on vacation for the last week.
To put the tail matter to a rest, it is simply a matter of how much mass it has, where it was tossed out into space (side of object), and what material it is. On an asteroid, it usually will remain perfectly stable and leave no tail, but if there was recently a collision, and it magically decided to stay in orbit, the particles -or tail, if you prefer- should either be pulled into the sun because they should not have the same mass and speed as the asteroid, or be thrown out infinitely into space for billions of years until they reach their next destiny. And an asteroid's "tail" would only last for a short time and are not even worth mentioning.
For a comet, you should expect the tail to fall behind it and then sink in towards the sun. (Remember these are particle we're talking about, they don't just trail and dissipate. Mass is never lost or gained.) And comets are generally made of any frozen material. For example, the most famous one is pluto.
The tail will NOT just float behind it save a one in a trillion possibility of a weird collision.
Note: Anything I told you is what I've learned from science channel that is highest probability. Also remember that this is an infinite universe.
-Supermj