If you're open to using a third-party library, the following will work using Java 8 with Eclipse Collections.
CharSet set = CharSets.immutable.with('a', 'b', 'c');
CharAdapter chars = Strings.asChars("a1b2c3");
String string = chars.select(set::contains).toString();
Assert.assertEquals("abc", string);
Eclipse Collections has support for primitive collections so there is no need to box char values as Character instances.
You can also exclude instead of include characters using the method named reject which is the opposite of select.
CharSet set = CharSets.immutable.with('a', 'b', 'c');
CharAdapter chars = Strings.asChars("a1b2c3");
String include = chars.select(set::contains).toString();
String exclude = chars.reject(set::contains).toString();
Assert.assertEquals("abc", include);
Assert.assertEquals("123", exclude);
Note: I am a committer for Eclipse Collections.