EDIT: It is actually even easier to use Type.IsInstanceOfType if you cannot make it a general method:
private bool CheckJumpObstacle(Type type) { return GameItems.Any(x =>type.IsInstanceOfType(x)); }
It looks like you probably want Type.IsAssignableFrom
if (go != null && type.IsAssignableFrom(go.GetType());
Note that this assumes that you want the inherited types to match.
Also, if at all possible, use generics instead. Among other things, this will make the method very simple:
private bool CheckJumpObstacle<T>() { return GameItems.OfType<T>().Any(); }
Even without this, you can still use LINQ to simplify it:
private bool CheckJumpObstacle(Type type) { return GameItems.Any(x => x != null && type.IsAssignableFrom(x.GetType())); }
Obviously, if you do not expect any null values, you can get rid of invalidation checking.
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