When should you stop chasing a piece? I'm only a beginner, and I've found that sometimes a piece of mine (esp Queen) will just get chased around relentlessly causing problems, so I've started trying to do the same, but I seem to sometimes overextend and it feels like I am "following" not chasing.  Any advice on when to stop chasing?  What should I be looking for when deciding to chase or not?

Particularly bad example (playing Black)

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1. f4 d5 2. d4 c5 3. Nf3 cxd4 4. Qxd4 Nc6 5. Qe3 d4 6. Qb3 Be6 7. Qxb7 Na5 8. Qb5+
If you chase an enemy piece, you should probably consider three aspects:


Basic tactical considerations.
Do you improve the position of the piece you use to chase the enemy piece?
Do you force the enemy piece on a worse or a better square?


In your example this means if you don't see that you are blundering pawns and pieces, all considerations of chasing or not chasing are pretty moot i.e. you have to think about whether you blunder something (like your b-pawn), whether your opponent blundered something (i.e. maybe you can already use his disorganised pieces for a tactical blow).

4…Nc6 was good, because you improve your knight and you chase the white queen away from a central square. 

5…d4 is already somewhat more ambiguous because the queen is pretty bad on e3 (blocks the e-pawn) and probably better on b3 (attacks b7).

6…Be6 blunders the b7 pawn, but maybe you could have continued Bd5 to set up even more attacks on the queen, like Rb8 or a knight move. This starts to look dangerous for white, because you are building up a big development lead.

7…Na5 puts your knight on a worse square (knights on the rim...), so you'd need a good reason to do so and apparently you didn't have one.