As the headaches are described as constant and debilitating , and no one has said it yet, maybe checking them out with a doctor might be a good idea. There is no real information to even guess at what is going on to cause the headaches. No information on age, health or if medical advice has been sought. Are they new?, does the person often get headaches? have they been diagnosed with any particular type of headache before etc.
If you are taking on the role of doctor and advising someone on treatment options with glp medications ( which are prescription medications for a reason ) you really need to be very certain that you understand what is going on, and if you do not know exactly what is going on to cause the headaches, then medical advice is required especially as you are talking about someone else. If it is yourself then you can decide for yourself what risks you are happy to accept , but if treating or advising someone else then I would think great care needs to be taken that you do not inadvertently cause harm. It is unlikely that there is a serious problem here but if you do not know that for certain then the only reasonable approach is to get expert advice.
The desire to share the tremendously useful effects these medications have with others is perfectly reasonable, but some degree of medical supervision is definitely preferable, and in the very unlikely event of someone having a fatal side effect, you could be liable if you provided the medication, and if you provide the medication or even just advice there is very definitely a duty of care, whether you are qualified to provide it or not.
In general tirzepatide is not known to make migraines worse and may even help but headache is listed as a side effect in about 5% of patients in the studies, and there is a large number of possible reasons it could make headaches more likely, including changes in diet and fluid intake. The simplest experiment to see if tirzepatide is responsible for the headaches, is to stop it and see if they go away, but given it's long half life it might take a few weeks. Once a explanation for the problem exists, and the person may well have a known and medically treated cause for the headaches, but there is no information here to know one way or the other, then it might be time to work out what alternative medication might be suitable.