The key is to write a custom appender. You don't say which unit testing framework you use, but for JUnit I needed to do something similar (it was a little more complex than just all errors, but basically the same concept), and created a JUnit @Rule that added my appender, and the appender fails the test as needed.
I place my code for this answer in the public domain:
import ch.qos.logback.classic.Level;
import ch.qos.logback.classic.Logger;
import ch.qos.logback.classic.LoggerContext;
import ch.qos.logback.classic.spi.ILoggingEvent;
import ch.qos.logback.core.AppenderBase;
import org.junit.rules.ExternalResource;
import org.slf4j.LoggerFactory;
import static org.junit.Assert.fail;
/**
 * A JUnit {@link org.junit.Rule} which attaches itself to Logback, and fails the test if an error is logged.
 * Designed for use in some tests, as if the system would log an error, that indicates that something
 * went wrong, even though the error was correctly caught and logged.
 */
public class FailOnErrorLogged extends ExternalResource {
    private FailOnErrorAppender appender;
    @Override
    protected void before() throws Throwable {
        super.before();
        final LoggerContext loggerContext = (LoggerContext)(LoggerFactory.getILoggerFactory());
        final Logger rootLogger = (Logger)(LoggerFactory.getLogger(Logger.ROOT_LOGGER_NAME));
        appender = new FailOnErrorAppender();
        appender.setContext(loggerContext);
        appender.start();
        rootLogger.addAppender(appender);
    }
    @Override
    protected void after() {
        appender.stop();
        final Logger rootLogger = (Logger)(LoggerFactory.getLogger(Logger.ROOT_LOGGER_NAME));
        rootLogger.detachAppender(appender);
        super.after();
    }
    private static class FailOnErrorAppender extends AppenderBase<ILoggingEvent> {
        @Override
        protected void append(final ILoggingEvent eventObject) {
            if (eventObject.getLevel().isGreaterOrEqual(Level.ERROR)) {
                fail("Error logged: " + eventObject.getFormattedMessage());
            }
        }
    }
}
An example of usage:
import org.junit.Rule;
import org.junit.Test;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
public class ExampleTest {
    private static final Logger log = LoggerFactory.getLogger(ExampleTest.class);
    @Rule
    public FailOnErrorLogged failOnErrorLogged = new FailOnErrorLogged();
    @Test
    public void testError() {
        log.error("Test Error");
    }
    @Test
    public void testInfo() {
        log.info("Test Info");
    }
}
The testError method fails and the testInfo method passes. It works the same if the test calls the real class-under-test that logs an error as well.