Get muratic acid from a pool store. Makes your pot just like new.

Unless your pot is absolutely foul with mineral deposits, I would leave them. Some of this is called "patina" and the look of age is appropriate for bonsai pots. Why would you put an ancient looking tree in a brand new looking pot? In fact, my teacher stores all his pots under his benches so they get the benefit of the runoff while he is watering! When we repot a tree and choose a pot, we simply scrub out the inside of the pot with a brush and some water.

Chris

Here is a link that might be useful: Great show!

For what it's worth, I don't think "patina" covers lime stains on newer pots.

The patina on old pots isn't due to lime stains or neglect in cleaning them. You can't remove patina from old pots anyway, at least not without considerable effort. The patina in really old pots occurs in the interaction of fertilizer/rain/ soil/etc. on the glaze itself and is not surface dirt. The interaction of the chemicals in each of those things with the glaze changes the color and "character" of the glaze dramatically and permanently.

However, such time worn patina can't be confused with simple lime stains on the bottom and sides of a new bonsai pot.It's not the same thing. Lime stains are simply an accretion of minerals on the surface of the pot and not "patina".

In order to remove them, you can use strong chemicals (the fast way), or by simply burying them in the ground or under something acidic, like pine mulch, sphagnum moss, compost, etc. for a year or so. I bury my pots in pine bark mulch for the winter. I have very little build up of lime scale on them, as the acidity of the mulch and microbes in it keep the stains to a minimum. This treatment also increases the "patina" of the pot, as the tannins and other stuff in the mulch alters the glazes over time. It's not much, but it can be a bit noticeable over a decade or so.

When using muratic acid to clean mineral deposits, do I then wash the pot after? Does the muratic acid have any effects on the next tree to be planted in the cleaned pot? In other words, is there any residue from the acid to be concerned about?

Well, yeah, it's an ACID that can dissolve minerals that can't be removed otherwise. Yes, wash the pot thoroughly after treating it and before using it for a plant.