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When I was younger I took a couple Harmonium classes and the teacher taught us how to play all of the scales. We weren't allowed to place our thumbs on the black notes.
I haven't had any piano instructions, so I don't know if that requirement is there as well, but I've noticed at concerts, recitals, and on Youtube, people generally play freely.
Why would the teacher ingrain the "No thumb on black notes" rule? Was there a purpose for this?
There's no hard rule to say that you shouldn't use your thumb on black notes - some pieces would be near-impossible to play without doing so!
However, there are some reasons why you might be better off by not doing so.
One is that the thumb is shorter than the rest of your digits, so it makes sense to reserve it for the easy-to-reach white keys. You specifically mention scales - so here, it does indeed make sense to reserve your thumb for the white notes, as you require fluid, step-by-step movement. Try crossing your thumb under a finger from a white to black key and see the awkward movement you make! The exception here is of course pieces in keys which make heavy use of the black keys, in which case you'll find using your thumb to be more convenient than not doing so.
Another situation is playing chords - you'll notice that using your thumb on a black key in many chords will cause you to twist your wrist slightly. Again, there are exceptions where using the thumb is more comfortable than not.
Another reason is historical - Prior to Bach and his contemporaries, conventional fingering avoided using thumbs altogether - black keys or otherwise! Again, there would be exceptions; e.g. for playing octaves or large intervals. If you wanted to be 'historically accurate' you might want to avoid your thumbs being used, but 'historically accurate' does not equal 'better'!
On the Indian harmonium, you are generally playing with one hand on a short keyboard (two or three octaves) and you are only playing one note at a time. So fingerings would not be so crucial. However, when you learn Western technique, the fingering patterns for piano are designed for two-handed playing across multiple octaves, and to facilitate rapid scales.
The main reason not to put your thumb on a black key is that when you do, it tends to be more difficult to move your fingers quickly to the next key or keys you need to strike. Experience will teach you this.
All the Western piano books provide exact fingering patterns for all twelve Western major and minor scales, in one hand and with both hands parallel, across two or more octaves. Due to the way that the keys are laid out on keyboard instruments, the fingering pattern for each of the traditional scales is distinctly different, and must be memorized.
Historical footnote (I'll leave it to you to look up a reference.) While the layout of the keyboard had been the same for centuries before, keyboard players rarely used their thumbs at all before J. S. Bach (1685-1750), who pioneered and popularized the technique of fingerings that involved the thumb.
Actually the rule applies only for your little finger and not thumb. The main reason why you are not allowed to use thumb is, you couldn't be able to generate speed while playing third or fourth kalas, as compared to index finger and others.
However in our school we follow the tradition of playing Harmonium with all fingers, what you need is only a good output and fingerings doesn't matter at all stage.
