There are two that annoy me quite a bit.
Date/Calendar
First, the Java Date and Calendar classes are seriously messed up. I know there are proposals to fix them, I just hope they succeed.
Calendar.get(Calendar.DAY_OF_MONTH) is 1-based
Calendar.get(Calendar.MONTH) is 0-based
Auto-boxing preventing thinking
The other one is Integer vs int (this goes for any primitive version of an object). This is specifically an annoyance caused by not thinking of Integer as different from int (since you can treat them the same much of the time due to auto-boxing).
int x = 5;
int y = 5;
Integer z = new Integer(5);
Integer t = new Integer(5);
System.out.println(5 == x); // Prints true
System.out.println(x == y); // Prints true
System.out.println(x == z); // Prints true (auto-boxing can be so nice)
System.out.println(5 == z); // Prints true
System.out.println(z == t); // Prints SOMETHING
Since z and t are objects, even they though hold the same value, they are (most likely) different objects. What you really meant is:
System.out.println(z.equals(t)); // Prints true
This one can be a pain to track down. You go debugging something, everything looks fine, and you finally end up finding that your problem is that 5 != 5 when both are objects.
Being able to say
List<Integer> stuff = new ArrayList<Integer>();
stuff.add(5);
is so nice. It made Java so much more usable to not have to put all those "new Integer(5)"s and "((Integer) list.get(3)).intValue()" lines all over the place. But those benefits come with this gotcha.