Short answer: not while it is active. And the reason is more specific than "it will hurt."
Why not
A cold sore is an open, infectious lesion. Herpes simplex reactivates, blisters form, they rupture, and the fluid contains active virus. That is the contagious stage.
Applying adhesive over it and peeling it off in the morning does several unhelpful things at once:
It spreads the virus across the area under the tape. Autoinoculation — moving virus from an active lesion to adjacent skin — is a genuine mechanism, and mouth tape covers precisely the perimeter where new sores would establish. You could turn one sore into several.
It breaks the lesion open. Cold sores heal by crusting. Peeling adhesive off a scab strips it, restarts the bleeding, prolongs healing and raises the risk of scarring and secondary bacterial infection.
It is painful, which is the least of it but will end the experiment anyway.
The tape is then contaminated. Single-use is not optional here.
What about the prodrome?
The tingling, itching or burning phase before anything visible — usually 12 to 24 hours ahead.
Stop taping at that point rather than waiting for a blister. It is the earliest and best moment to start antiviral treatment, and it is also the point at which the area is about to become an open lesion. Continuing to tape through the prodrome is how you get an outbreak under adhesive.
When you can start again
Once the sore has fully crusted and the crust has come away on its own, leaving intact skin. Not while a scab is still present.
For most outbreaks that is somewhere between seven and twelve days, though antiviral treatment started early shortens it. Do not rush the last couple of days for the sake of one more night of taping — a scab pulled off at day eight sets you back a week.
Could taping trigger one?
Fair question, and the answer is: probably not directly, but the surrounding conditions matter.
Established triggers are UV exposure, fever and illness, physical or emotional stress, hormonal changes, and local trauma to the lip. Adhesive on healthy skin is not on that list.
But local trauma is. Repeated aggressive removal, dragging tape off dry, and adhesive that irritates the vermilion border are all minor trauma to exactly the region where cold sores establish. If you get frequent outbreaks and you tape nightly, that is worth considering.
Two things reduce it. Remove the tape wet — in the shower or with a warm damp flannel — and peel low and slow rather than pulling outward. (Proper removal technique.) And place the strip across the lip seam with a margin of skin above and below, rather than centring it on the lip tissue itself.
Adhesive choice matters here more than average. A gentle, hypoallergenic medical-grade adhesive is doing less to that skin over 300 nights a year than an aggressive one. Titan's bamboo silk tape uses an adhesive that has been through ISO 10993 biocompatibility testing — cytotoxicity, sensitization and irritation, scoring 0.0 out of 8 for irritation. That testing was on the adhesive and was commissioned by its manufacturer, Henkel, rather than by Titan. (The reports.)
What is definitely worse: athletic tape, wound-closure tape, or anything not formulated for facial skin. (Backings and adhesives compared.)
What to do on those nights
You still want nasal breathing; you just cannot tape for it.
Open the nose instead. A nasal strip sits on the bridge, nowhere near the lips, so it is entirely unaffected. TitanAir strips lift the nasal valve with spring-loaded bands. (Where it goes.)
Side sleeping rather than on your back keeps the jaw and tongue better positioned.
Elevate your head, treat any congestion, humidify a dry room.
Keep practising nasal breathing while awake, which is most of the work anyway. (The daytime habit.)
Ten nights off does not undo weeks of retraining.
The related cases
Same reasoning applies to any broken skin around the mouth: angular cheilitis, impetigo, eczema flares, chapped and split lips, a healing piercing, or recent dental or oral surgery. If the skin is not intact, do not tape it.
The bottom line
Not while it is active, and stop at the first tingle rather than waiting for the blister. Adhesive over an open lesion can spread the virus across the exact area the tape covers, and peeling a scab restarts the healing clock.
Wait until the crust has come away on its own. Use a nasal strip and side sleeping in the meantime.