Yes — zinc and copper are commonly taken together, and copper is in fact added to high-dose zinc supplements on purpose to prevent zinc from causing copper deficiency, but stick to sensible doses and ask your pharmacist before taking high-dose zinc long-term.
The concern with zinc isn't that it clashes with copper but the reverse: high doses of zinc reduce how much copper your body can absorb, and over time this can cause copper deficiency, which the NHS notes can lead to anaemia and weakening of the bones. Taking copper alongside zinc counteracts this — which is exactly why the NIH's AREDS2 eye-health formula pairs 80 mg of zinc with 2 mg of copper specifically "to reduce the risk of copper deficiency anemia, a condition associated with high levels of zinc intake." (That 80 mg is a clinician-studied therapeutic dose well above the usual supplement range, not a casual recommendation.) So the two are complementary rather than a dangerous combination; the actual danger is high-dose zinc taken alone, without copper, for a long time.
Doses matter most. Most adults get enough zinc and copper from a varied diet (NHS: roughly 9.5 mg/day zinc for men, 7 mg for women; 1.2 mg/day copper). The NHS advises not taking more than 25 mg/day of zinc supplements unless a doctor tells you to, and keeping total copper at 10 mg/day or less. Signs that high zinc may be depleting your copper include unusual tiredness, anaemia, easy infections, numbness or weakness in the hands and feet, and loss of coordination/balance — the NIH notes copper-deficiency signs have occurred in people taking about 150 mg/day or more of zinc for up to two years. Avoid high-dose zinc on your own; if a clinician puts you on it, expect blood tests to monitor copper. Talk to a pharmacist or doctor before combining supplements if you are pregnant, have Wilson's disease or another copper-metabolism disorder, take other medicines, or use a multivitamin (to avoid doubling up). Call a professional if you develop persistent fatigue, numbness, or balance problems while taking zinc.
This is general reference, not medical advice, and not a guarantee of safety. Interactions depend on your doses, health conditions, and other medicines. Always confirm with your pharmacist or doctor before combining products, and follow the dosing on each label.