Perhaps this is a non-answer, but I wanted to state that there's very little musical reason, in my opinion, for that rule; it really sounds like a rule that will prevent you from making other errors (like doubling the leading tone, which is the third of a major triad).
Furthermore, I wanted to suggest that you treat this as a very bendable rule that can be broken.
Indeed, when you get to V–VI deceptive cadences in a minor key, you must double the third of the root-position VI chord (!), because it's better than doubling the root (which would be a result of parallel perfect octaves or an augmented second) or the fifth (which would result in parallel perfect fifths).
In other words, there are situations where this rule must be broken, so (in my opinion), don't worry too much about breaking it.
music.stackexchange.com/users/21766/Richard
One reason why may be to avoid parallel octaves.
Dominant chords are always major chords (or at least based on them), and doubling the third of a dominant chord means doubling the leading tone.
In common practice period voice leading, the leading tone should resolve to the tonic, so you get parallel octaves right there.
The common story I've read is that parallel octaves should be avoided because they make contrapuntal voices sound less independent from each other.
In practice, due to how rare they are, they also stick out to my ears in polyphonic writing in a rather raw way.
music.stackexchange.com/users/37354/Dekkadeci
As mentioned in Dekkadeci's answer, doubling the third can lead to using parallel octaves (neither of which sound bad) and it can sound like a voice dropped out.
This is a problem with the dominant chord; the third is note 7 of the scale and it strongly leads to the tonic.
(Parallel octaves or fifths can sound "thin" in harmonic texture.)
Another problem is that a major with a double third sounds like a Neapolitan Sixth (in perhaps another key).
It's a special sound that listeners are used to hearing in particular situations.
Neither of these is a risk with minor chord third doubling.
Doubling the third of a minor chord (in a major key) doubles the 1, 4, or 5 note.
This strengthens the sense of key.
(I'm not sure this is all that important though.)
music.stackexchange.com/users/23838/ttw
Although the chorale rules evolved somewhat organically, there may be a physics component to it as well. Consider the harmonic series for the note
C:
C, G, C, E, G, (7),
C.Now, the harmonic series for the note
E: E, B, E, G#, B, (7), E.And the harmonic series for the note
Eb:
Eb, Bb, Eb, G, Bb, (7)
, Eb.
In a C Major chord, a doubled E would also reinforce a G# against the chord's G -- Eew.
But in a C Minor chord, a doubled Eb would reinforce a G natural -- no problem.
music.stackexchange.com/users/53167/Tom Williams
In a major chord, the 3rd is probably the most 'active' note.
If the chord has a dominant function to the one following (as is so often the case in Common Practice harmony)
it will be particularly active, having a leading note function, and as @Dekkadeci says allowing two instances of it to resolve will give parallel octaves.
(The same reasoning applies to not doubling the 7th in dominant 7th shape chords.)
A minor 3rd is less active.
Doubling
it is as innocuous as doubling the 5th.
But, as always, use this 'rule' as a warning to listen carefully if you are tempted to double a major 3rd.
DOES
it stand out too much?
DOES
it cause bad part-leading?
music.stackexchange.com/users/17003/Laurence Payne
I think it has more to do with what we're used to hearing, culturally.
Most of the time, if we have 4 voices but only 3 notes, we'll double the root.
It sounds more stable and we perceive the chord's function more accurately.
Besides the voice leading perspective (avoiding parallel octaves/5ths)
we don't mind hearing 'C' in a C Major triad more strongly than the other notes, so the root ends up being doubled for aesthetic reasons (like in a rock 'n' roll band where the bass almost always plays the root of a chord).
I suspect cultural exposure has more influence on aesthetics than the overtone series.
music.stackexchange.com/users/53154/Nikki
The minor third has a much easier tone that the Major third, the major third is a much more dominant interval than the minor third.
it is also worth noting that one of the Major chords in tonal harmony (The Dominant) has the Leading tone note for a third, something which is a big no-no to double.
In general, the Major chords almost always want it root note doubled, sometimes you are by necessity forced to double the fifth
but you only do this when there is a good reason to do so.
music.stackexchange.com/users/7306/Neil Meyer
Dieter da la Motte says in his "Harmonielehre" Which tone is doubled in the sixth chord?
Ask ten textbooks.
They give ten different answers between the extremes of Bumcke ("The third should not be doubled") and Moser ("so that in the sixth chord all three doubling possibilities become more equal").
There is a lot more to say about doubling the 3rd in root position that I will post here later ...
music.stackexchange.com/users/56219/Albrecht Hügli
In a manual for four-part harmony that I am reading, it is suggested to avoid doubling the 3rd of a root position major chord, as it sounds harsh. On a minor root position chord however, it is acceptable.   Why is it better to double the 3rd of minor chords and less good in major chords? The overtone series has a tone very close to the major 3rd, so shouldn't doubling the major 3rd actually be preferable to doubling the minor 3rd?
Doubling the third of a root position major chord in four-part harmony
music.stackexchange.com
<harmony><voice-leading><satb>
music.stackexchange.com/questions/74990
music.stackexchange.com/users/53143/Stefanos
Dominant chords are always major chords (or at least based on them), and doubling the third of a dominant chord means doubling the leading tone. In common practice period voice leading, the leading tone should resolve to the tonic, so you get parallel octaves right there. It is also worth noting that one of the Major chords in tonal harmony (The Dominant) has the Leading tone note for a third, something which is a big no-no to double. In general, the Major chords almost always want its root note doubled, sometimes you are by necessity forced to double the fifth Major third is a much more dominant interval than the minor third. 
Dominant chords are always major chords (or at least based on them), and doubling a third of a dominant chord means doubling the leading tone. In common practice period voice leading, the leadingtone should resolve to the tonic, so you get parallel octaves right there. Major chords almost always want it root note doubled. Sometimes you are by necessity forced to double the fifth. Major third is a much more dominant interval than the minor third. One of the Major chords in tonal harmony (The Dominant) has the Leading tone note for a third.