Does this formula hold true for a source sheet covering a closed body of arbitrary shape?
If a streamline is closed, then there's no mass flow that can go throught it. That means that inside the streamline the net mass flow produced is null. You can have for example sinks and sources inside a closed streamline, but they must compensate each other to make the streamline remain closed.

Another important aspect about source sheets is that they actually don't match with the border of the body, i.e, with the closed streamline. If they did, then some flow produced by the sources would go out of the source sheet, and some other flow would be in. The source sheet needs to be inside the border of the body. Anyway you could find a source distribution that makes the source sheet to almost match the border of the body.

What I wanted to say, is that if the streamline is closed, then the sum of sources and sinks inside the body is null. If the sinks and sources are the source sheet itself, then this proposition is equivalent to: 

$$\oint_{Source\hspace{1pt}sheet} \lambda(s)ds=0\hspace{40pt}(1)$$ 

Note that each infinitesimal element of sheet, $ds$, produces an infinitesimal source strength of $\lambda(s)ds$ in the $s$ position. The relation $(1)$ means that the sum of all infinitesimal source strengths along the source sheet is null, or, in other words, that the net source strength is null.