I broke my hand and had surgery to replace my bone with a metal plate. What are the long-term side effects?

Replace? Usually a metal plate is used to hold the bone while it heals. The bone will eventually be strong without limitations. The plate is not removed unless it causes problems. If the plate was used to replace the bone (unlikely) then this is temporary and you will need more surgery.
Hardware removal. If you are talking about the hardware not the fracture then prominent hardware necessitating later removal is the most common problem. However even this does not occur with a high frequency. Soft tissue irratation is another potential problem.