Well, this algorithm can be at least improved by adding option to check for any list characters at position 0 and 1.

public class DeFront {

    //if character is one of these, it will be removed
    private static char[] firstLetterExcluded = "a".toCharArray();
    private static char[] secondLetterExcluded = "bcdef".toCharArray();

    public static String deFront(String str) {
        boolean removeFirst = false, removeSecond = false;

        for (char c : firstLetterExcluded) {
            if (str.charAt(0) == c) {
                removeFirst = true;
                break;
            }
        }

        for (char c : secondLetterExcluded) {
            if (str.charAt(1) == c) {
                removeFirst = true;
                break;
            }
        }
        //Before calling the stripping method, it's better to check if we need to
        if (removeFirst || removeSecond) {
            return stripString(str, removeFirst, removeSecond);
        } else {
            return str;
        }
    }

    private static String stripString(String str, boolean removeFirst, boolean removeSecond) {

        if (removeFirst && removeSecond) {
            return str.substring(2);
        } else if (removeFirst) {
            return str.substring(1);
        } else {
            //Using Stringbuilder takes more time
            return str.charAt(0) + str.substring(2);
        }


    }

}

Well, this algorithm can be at least improved by adding option to check for any list characters at position 0 and 1.

public class DeFront {

    //if character is one of these, it will be removed
    private static char[] firstLetterExcluded = "a".toCharArray();
    private static char[] secondLetterExcluded = "bcdef".toCharArray();

    public static String deFront(String str) {
        boolean removeFirst = false, removeSecond = false;

        for (char c : firstLetterExcluded) {
            if (str.charAt(0) == c) {
                removeFirst = true;
                break;
            }
        }

        for (char c : secondLetterExcluded) {
            if (str.charAt(1) == c) {
                removeSecond = true;
                break;
            }
        }
        //Before calling the stripping method, it's better to check if we need to
        if (removeFirst || removeSecond) {
            return stripString(str, removeFirst, removeSecond);
        } else {
            return str;
        }
    }

    private static String stripString(String str, boolean removeFirst, boolean removeSecond) {

        if (removeFirst && removeSecond) {
            return str.substring(2);
        } else if (removeFirst) {
            return str.substring(1);
        } else {
            //Using Stringbuilder takes more time
            return str.charAt(0) + str.substring(2);
        }


    }

}