Whether dogs can eat mushrooms safely depends on the type of mushroom. Here are some guidelines:
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Buttons/common white mushrooms (Agaricus bisporus): Generally safe in small amounts if cooked thoroughly. Provide high quality protein and fiber.
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Shiitake mushrooms: Safe cooked or dried in moderation. Provide B vitamins.
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Oyster mushrooms: Safe cooked or dried. Low calorie source of protein, fiber, vitamins.
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Portobello mushrooms: Safe cooked. Low calorie, high fiber vegetarian protein option.
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Enoki mushrooms: Generally safe cooked but may cause slight gastrointestinal upset in some dogs.
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Chanterelle mushrooms: Considered non-toxic but may cause slight GI upset in sensitive dogs. Cook thoroughly.
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Morels: Typically safe if cooked well. May be slightly irritating to the GI tract if eaten raw.
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Champignon mushrooms: Safe cooked or dried in moderation. Good source of fiber, vitamins.
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Eryngii (king oyster): Safe cooked or dried. Low calorie protein, fiber, minerals.
Some mushrooms to avoid include wild/unidentified varieties, as well as AMANITAS (death cap, destroying angel, false morel), which can be fatally toxic to dogs. 
Avoid mushroom supplements as well. Cook thoroughly and monitor your dog for any issues.