"In the four-part writing of diatonic seventh chords the fifth may be omitted, in which case the doubled note in a four-part texture should not be a tendency tone. The most common doubling is the root." (in Harmony text) What is a useful definition for a tendency tone in Harmony theory ?
In diatonic harmony, tendency tones tend to be one of two things:
- The leading tone, which has a tendency to resolve up to tonic (especially when in an outer voice), and
- Chordal sevenths, which have a tendency to resolve down by step (no matter what voice they're in).
The logic is as follows: if you double a tendency tone, since these tendency tones have a tendency to resolve in a particular way, doubling a tendency tone will result in parallel octaves, which is of course one of the most strident errors in four-part voice leading.
In a dominant 7th - tonic progression, the notes that form the tritone in the dom7, the 3rd and 7th of the chord, have a strong tendency to resolve to the root and 3rd of the subsequent tonic. (This desire of a dominant to reach a tonic is the harmonic basis of the whole Common Practice thing.)
We avoid doubling these notes because their active nature would predominate too much. And, as @Richard said, because if both doubled instances resolved 'correctly' it would create parallel octaves.