So yeah, you can make a prosthetic that gives the appearance of the mouth sewn shut. It isn't super hard. You build up layers of liquid latex on a sheet of glass, and then, once thick enough, powder it heavily so it doesn't stick to itself, and loosely sew it shut with a leather-needle and upholstery thread.
If using platsil gel-10 silicone, you can just apply the tinted silicone directly to the face with a popsicle stick, and once cured and any finishing coloring touches are applied, remove the appliances (for safety), stitch them together and reapply with a thin layer of more silicone.
But as said, this isn't a very con friendly technique. You can improve things by making the stitching loose enough to be able to talk, or making the stitching improperly so that it can easily be completely loosed and pushed to the corners of the mouth when needed, and then tightened again later. But for that, you'll need a good length of slack on the end, or you'd have to resew it every time. This would probably only work for the silicone, as the latex would present more friction on the thread.
You can peel either prosthetic off without much difficulty, and reapply it; but every time you do, particularly with latex, it will get a little crummier looking.
I'm not being serious here, but you could get someone skilled at body-modification to actually sew your lips shut. It'd hurt like mad, and you'd run a serious risk of infection, but yeah...one removed, the holes would close up...eventually. Seriously...this is obviously stupid, don't do this.