When you play units under the effect of Fanfare, do you draw once for each unit played up to a maximum of three draws, or do you draw once after playing three units total?

Each unit gets its own draw — Fanfare reads "the next three times you play a unit this turn, draw 1," so it's three separate draw events, one per unit played, not one lump draw after the third unit.

So playing three units this turn = three draws (one each), and playing only two = two draws. If a single card plays two units at once, those still count as two separate "play a unit" events, so you'd get two draws from them (FAQ #12756) — the triggers go on the chain one at a time. Once the third unit has been played the effect is used up, so a fourth unit this turn draws nothing.