You can actually extend Consumer (and Function etc.) with a new interface that handles exceptions -- using Java 8's default methods!

Consider this interface (extends Consumer):

@FunctionalInterface
public interface ThrowingConsumer<T> extends Consumer<T> {

    @Override
    default void accept(final T elem) {
        try {
            acceptThrows(elem);
        } catch (final Exception e) {
            // Implement your own exception handling logic here..
            // For example:
            System.out.println("handling an exception...");
            // Or ...
            throw new RuntimeException(e);
        }
    }

    void acceptThrows(T elem) throws Exception;

}

Then, for example, if you have a list:

final List<String> list = Arrays.asList("A", "B", "C");

If you want to consume it (eg. with forEach) with some code that throws exceptions, you would traditionally have set up a try/catch block:

final Consumer<String> consumer = aps -> {
    try {
        // maybe some other code here...
        throw new Exception("asdas");
    } catch (final Exception ex) {
        System.out.println("handling an exception...");
    }
};
list.forEach(consumer);

But with this new interface, you can instantiate it with a lambda expression and the compiler will not complain:

final ThrowingConsumer<String> throwingConsumer = aps -> {
    // maybe some other code here...
    throw new Exception("asdas");
};
list.forEach(throwingConsumer);

Or even just cast it to be more succinct!:

list.forEach((ThrowingConsumer<String>) aps -> {
    // maybe some other code here...
    throw new Exception("asda");
});

Update

Looks like there's a very nice utility library part of Durian called Errors which can be used to solve this problem with a lot more flexibility. For example, in my implementation above I've explicitly defined the error handling policy (System.out... or throw RuntimeException), whereas Durian's Errors allow you to apply a policy on the fly via a large suite of utility methods. Thanks for sharing it, @NedTwigg!.

Sample usage:

list.forEach(Errors.rethrow().wrap(c -> somethingThatThrows(c)));

You can throw exceptions in lambdas.

A lambda is allowed to throw the same exceptions as the functional interface implemented by the lambda.

If the method of the functional interface doesn't have a throws clause, the lambda can't throw CheckedExceptions. (You still can throw RuntimeExceptions).

In your particular case, Map.forEach uses a BiConsumer as a parameter, BiConsumer is defined as:

public interface BiConsumer<T, U> {
    void accept(T t, U u);
}

A lambda expression for this functional interface can't throw CheckedExceptions.

The methods in the functional interfaces defined in java.util.function package don't throw exceptions, but you can use other functional interfaces or create your own to be able to throw exceptions, i.e. given this interface:

public interface MyFunctionalInterface<T> {
    void accept(T t) throws Exception;
}

The following code would be legal:

MyFunctionalInterface<String> f = (s)->throw new Exception("Exception thrown in lambda");