Okay, here's the deal: dataContext.Table1s is of type IQueryable<T>. IQueryable<T> defines Where and Any methods that take a predicate of type Expression<Func<T, bool>>. The Expression<> wrapper is critical, as this is what allows LINQ to SQL to translate your lambda expression to SQL and execute it on the database server.
However, IQueryable<T> also includes IEnumerable<T>. IEnumerable<T> also defines Where and Any methods, but the IEnumerable version takes a predicate of type Func<T, bool>. Because this is a compiled function and not an expression, it can't be translated to SQL. As a result, this code...
Func<Table1, bool> lambda = x => x.Id > 1000;
var result = dataContext.Table1s.Where(lambda);
...will pull EVERY record out of Table1s into memory, and then filter the records in memory. It works, but it's really bad news if your table is large.
Func<Table1, bool> lambda = x => x.Id > 1000;
var result = dataContext.Table2s.Where(x => x.Table1s.Any(lambda));
This version has two lambda expressions. The second one, being passed directly into Where, is an Expression that includes a reference to a Func. You can't mix the two, and the error message you're getting is telling you that the call to Any is expecting an Expression but you're passing in a Func.
var result = dataContext.Table2s.Where(x => x.Table1s.Any(y => y.Id > 1000));
In this version, your inner lambda is automatically being converted to an Expression because that's the only choice if you want your code to be transformed into SQL by LINQ to SQL. In the other cases, you're forcing the lambda to be a Func instead of an Expression - in this case you're not, so it works.
What's the solution? It's actually pretty simple:
Expression<Func<Table1, bool>> lambda = x => x.Id > 1000;
var lamda = x => x.Id > 1000;. Don't know it'll help, but it might...Func<>orExpression<Func<>>and in your example the compiler won't be able to tell which one you want, and will throw an error.