I hurt my arm lifting heavy boxes. My forearm is sore and my hand is tingling. Is rest and ice enough for treatment? Should I get a brace?

May need more. Rest and brace are ok, but the fact that you have tingling in your hands may mean that their is a nerve involve, i would suggest that you get further evaluation.
May or may not. Try ice and rest as well as brace. After 2 or 3 days use heat and brace. If symptoms do not resolve after 7-10 days then orthopedic and or neurological evaluation is warranted.