There are a number of ways to approach answering this question and the most unnerving (no pun intended) aspect of it is that it is not explored how deep a range of emotions we are seeking to emulate. The very fact that it is claimed that all human movement can be simulated through the use of control bars implies that there is some simplification at work; can this puppet be manipulated in such a way that it can accurately play Chopin while seated at a piano?
How much extra effort would the puppeteer have to undertake by comparison to having someone perform it directly? One could argue that having a puppet emulate Chopin would take the puppeteer far more skill than it would take to play Chopin directly, and that's saying something.
So, back to facial expressions. If we think the fine motor skills of the human hand are tough to emulate with strings, then the subtleties of human expression would also be quite difficult. That said, let's go through the mechanics of it all.
There are 43 muscles in the human face, and these are largely controlled via 5 branches of a primary nerve line. Sure, these muscles do more than just demonstrate emotion; they're also used to blink eyes, chew food, etc. But for the purposes of this answer let's assume that we need all 43 to demonstrate emotions.
We do have some economies here. Most emotions involve using muscles in our face symmetrically, and we've already stated that there really are only 5 major branches of nerves that drive the face. Assuming therefore that we an limit the strings to around 10 (2 for each nerve line), it's possible to emulate some emotions via marionette control bars.
That said, unlike the fingers, you can't assume that gravity is your friend. When operating the fingers to play a piano, you can merely drop the string a little on the finger that needs to strike the key, and raise it again when the finger needs to come up. With facial muscles, it's not that simple. You might be able to do something similar with eyebrows, perhaps even with the nose to simulate distaste (turning up your nose), but the lips for example will need far more control, including lateral movements. Surprise will need you pushing the edges of the lips together, while the top lip goes up, the bottom lip goes down to form a crude 'O' with the mouth. Sadness needs the two edges to go down, but the middle of both lips to remain where they are. This is not as easy to manage without looking at emulating musculature rather than areas of the face like the mouth directly with strings.
If you emulate the musculature, you'll be capable of a wide range of facial expressions, but at the cost of a highly complex set of controls. Could a single person use them? I doubt it, but then I also doubt that a single person could control a marionette with sufficient precision and skill to play Chopin, especially given the complexity of control that would surround the hands alone.
The shortest answer I can give you is that the complexity of your control system directly correlates with the range and detail of emotional display possible. I suspect that your marionette's controls will fall somewhere within a happy balance between the two.