What is a good way to build harmonic tension in music without adding too much dissonance?

Judicious use of minor chords, diminished triads or diminished seventh and half-diminished 7th chords would probably achieve the desired effect as would a diminished 7th chord that moves upward by changing position or a chromatic succession of diminished chords. Augmented chords would give a feeling of tension and unrest.

A rapidly repeated note or chord can build tension especially if placed over a chromatically rising bass line.

You can even use more dissonant chords if the dissonant notes are spaced further apart with other notes between them. The level of dissonance can also be controlled(lessened) by placing the dissonant notes in different instrumental timbres or registers.

Musician, composer, teacher.

Study Wagner. You'll have to do it with time. The longer you don't resolve chords, the more tension you build. Listen to Wagner's "Tristan und Isolde." There's not a V-I cadence until the end of the 4 hours of opera.

dissonances are good. they don't disturb anything if you know how to put them in a music. choose the classic way I-IV-V or anything else that sounds familiar with I-IV-V but sound more interesting, not classical at all. the difference is in tension.