Are there any problems with zinc collecting in the body over time?

Yes w/ every vitamin. You can get too much of anything, even water. While in the short term certain supplements may be helpful, you will correct a deficiency eventually. At that point, continuing to take the supplement at the same rate will cause symptoms. If it's water, you wake up at night to urinate. If it's vitamin c, you'll probably get heartburn. Will zinc it may be severe nausea and vomiting.
Yes. Zinc intoxication coul be from zinc chloride used in the pillow manufacturing industry zinc poisoning. A condition caused by industrial exposure or by excess ingestion of zinc supplements. Gives:headache, n&v, dehydration, stomach ache, incoordination, renal damage, immune deficiency mcgraw-hill concise dictionary of modern medicine. © 2002 by the mcgraw-hill companies, inc.