Lots of people have been taking supplemental zinc to boost their immune systems during the pandemic. It’s possible to get too much:
NIH – Zinc Consumer Fact Sheet
Signs of too much zinc include nausea, vomiting, loss of appetite, stomach cramps, diarrhea, and headaches. When people take too much zinc for a long time, they sometimes have problems such as low copper levels, lower immunity, and low levels of HDL cholesterol (the “good” cholesterol).
NIH – Zinc Health Professional Fact Sheet
Reductions in a copper-containing enzyme, a marker of copper status, have been reported with even moderately high zinc intakes of approximately 60 mg/day for up to 10 weeks.
The doses of zinc used in the AREDS [Age-Related Eye Disease Study] study (80 mg per day of zinc in the form of zinc oxide for 6.3 years, on average) have been associated with a significant increase in hospitalizations for genitourinary causes, raising the possibility that chronically high intakes of zinc adversely affect some aspects of urinary physiology .
What is high for you may not be high for someone else. Glancing at the RDA’s, anything over 11 mg/day doesn’t appear necessary and may be harmful – especially if taken repeatedly.
Fanatic Cook 
The conventional wisdom on this has long been to supplement a bit of copper as well. One site says, “Many alternative and integrative practitioners recommend a ratio of 15 mg of zinc to 1 mg of copper”. Some zinc supplements also contain copper for this reason.
I think an interesting question is whether you can get too much zinc if you’re also getting a proportional amount of copper. (Presumably there must be a toxic dose, but is it small enough to happen much in real life?)
I’ll end with this very interesting medical case study from the NY Times about a patient whose excess supplemental zinc caused a severe copper deficiency: https://www.nytimes.com/2006/11/12/magazine/12wwln_diagnosis.html
A quote: “zinc and copper were absorbed by the body through the same cellular doorway. If you consumed too much zinc, you absorbed too little copper.”
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