If a player uses disguise self to change their appearance, would you let them speak but have the illusion created by the spell appear as though their mouth/head/lips weren't moving?

I wouldn't, no.

RAW, it isn't a function of the spell, imo.

I suppose it depends. If you make yourself look like a hairy beast like Cousin It, or if you create a hat or cloak with a really large cowl/brim then it could hide your face. 

But your lips would still move.

Thrikreen are telepathic so they don't have to move their mouth parts to talk.

Quote from FramesJanko >>

If a player uses disguise self to change their appearance, would you let them speak but have the illusion created by the spell appear as though their mouth/head/lips weren't moving?

Well as you wrote it, not really, as a DM I would help the player achieve what they want. 

RAW:

You make yourself—including your clothing, armor, weapons, and other belongings on your person—look different until the spell ends. You can seem 1 foot shorter or taller and can appear heavier or lighter. You must adopt a form that has the same basic arrangement of limbs as you have. Otherwise, the extent of the illusion is up to you.

I would point out a film where the main character's superpower was basically this. 

And let them know the joys of hiding a face with clothing. 

if you did disguise self where the illusion showed your mouth being sewn shut  yes  otherwise no however  why not use  performance instead?

Quote from FramesJanko >>

If a player uses disguise self to change their appearance, would you let them speak but have the illusion created by the spell appear as though their mouth/head/lips weren't moving?

 What does disguise self have to do with your lips? I guess that would be my first question to the player. 

Whenever a player asks me about some seemingly random and odd detail, I have the same reaction, the same question comes to mind.

Why aren't you telling me what it is that you want to do? 

Like, instead of asking me twenty questions about the kind of metal fastenings on that chandelier, what the melting point is, what kind of heat a spell puts out etc...why aren't you asking if Fire Bolt would be enough to bring that down?

I'll tell you why. It's because you think that if you just ask me straight what you want to do, I'll say no, so you're trying to sneak it past me. Ironically, I'm actually quite likely to agree to something cool, even if I put in restraints to stop it being used all the time. If you try to sneak it past me, I'm likely to say no just so I don't get unintended consequences.

* If you're a player, just ask your DM straight about what your plans are. They're probably like me and more likely to agree of they know where this is going and can therefore make it work without breaking the game and making it boring.

* If you're a DM and a player asked this, then insist they be open and honest. You don't want to be opening Pandora's Box when you try to realise their vision.