Some foods are bad:
Excess of some dark greens (spinach, dandelion, broccoli stem & kale for example) can leach calcium from your finch's body. They contain oxalates that can bind to calcium and prevent it from being absorbed by the body. While these foods are actually high in calcium themselves they can do more harm than good if given to often. Cooking these foods before giving them to the finches will help decrease the effect of the oxalates.
Other bad foods are:
Alcohol & alcoholic beverages
Avocado (skin, pit)
Asparagus
Caffeine
Chocolate
Coffee
Corn (stored in humid area)
Eggplant
Highly salted foods
Milk Products (birds can not digest lactose)
Mushrooms
Olives
Onion
Rhubarb
Seeds (in quantity apple, apricot, bitter almond)
Seeds (in quantity cherry, peach, pear, plum)
Soft Drinks (with caffeine or colas)
Spinach
Tea
Foods that are good:
Remember, too much of any one thing is always bad. If your finch doesn't seem to like something, try something else. Just like people, individual finches will have food preferences. Always keep seeds and fresh water available. It's also very important to remove fresh foods from the cage each day before they spoil. Finches are just as likely to get sick from spoiled or moldy food as you would.
Apple slices
Apricot
Banana
Bean Sprouts (Mung and Fenugreek)
Blackberry
Bulgur Wheat
Cabbage
Cauliflower
Carrot (thin slices, no greens)
Cherry
Egg: Hard Boiled (with shell)
Grape slices
Green Beans (cooked & mashed)
Guava
Mango
Melons
Natural yogurt (sweetened with fresh fruit)
Nectarine
Oats
Orange
Peach
Peas (cooked & mashed)
Pear
Pearl barley
Pineapple
Potatoes (no skin & boiled)
Pumpkin (seeds as well)
Rice: White or Brown (cooked)
Sprouts (sprout your own finch seed or millet)
Strawberry
Sweet Potato (boiled)
Raspberry
http://www.finches.org/f-finchfood.mgi