I'm a Wizard with 20 Intelligence, who has obviously memorized a few songs. Could I use my Improved Minor Illusion (Illusion School) to conjure up an illusory flute, and make sounds that make it seem like I'm actually playing it?
This would be for the purpose of securing free room & board at taverns, etc.
You can create an image of a flute, but when you try to "play" it, the illusion reveals:
Physical interaction with the object reveals it to be an illusion, because things can pass through it.
You actually can play a melody using the basic Minor Illusion cantrip:
If you create a sound, its volume can range from a whisper to a scream. It can be your voice, someone else's voice, a lion's roar, a beating of drums, or any other sound you choose.
It should be fine to "play the invisible flute" for the purposes of entertainment. You can also pretend you are playing a real flute (or a flute-like stick) while casting the cantrip.
Minor Illusion only lasts one minute, so there would be gaps in your song. And anyone who was listening critically would probably get a save.
If you were using your illusion skills to entertain an audience (e.g. to earn money), that would require a Charisma (Performance) check.
Minor Illusion says about the sounds it can create:
It can be (...) any other sound you choose. (...) you can make discrete sounds at different times before the spell ends.
So creating the illusion of music for the duration of one minute should not be a problem. The spell further says about illusory objects, in your case, a flute:
Physical interaction with the object reveals it to be an illusion, because things can pass through it.
That need not mean you cannot fake playing an illusionary flute, but it will make it more difficult. Your viewers are not interacting with the illusion, you are, and you can try to deceive them to avoid giving its nature away. This will be hard as the flute is not to be moving like it normally would when you play a flute.
The real challenge however is that high intelligence does typcially not translate into being good at performance. Performance as a skill is described in the PHB, p. 180:
Performance. Your Charisma (Performance) check determines how well you can delight an audience with music, dance, acting, storytelling, or some other form of entertainment.
Even with illusionary music, you will need to put on a performance. In addition, you need to fake playing a flute and succeed at deception (PHB p. 179):
Deception. Your Charisma (Deception) check determines whether you can convincingly hide the truth, either verbally or through your actions.
At that point, you could just as well make it a magic show with fantastic illusions, and save yourself the pain of trying to fake playing a flute to begin with.
P.S. For comparison, the Entertainer background (p. 130 PHB) has a feature called "By Popular Demand" that allows them to obtain such free room and board.
You can always find a place to perform, usually in an inn or tavern (...). At such a place, you receive free lodging and food of a modest or comfortable standard (depending on the quality of the establishment).
So also from a game fairness perspective, it should not be possible to get this benefit "for free", without succeeding at some kind of challenge.
