When using the Source Panel method and dealing only with sources and sinks, why do lift and drag equal zero?
A single source, or a source sheet comprised of infinitesimal sources (sinks are just sources with negative strength), produces a velocity field that contains no circulation. Without circulation, it can't possibly generate any lift. The Source Panel Method is just a discretized computational method for source sheet. It, therefore, only produces non-lifting flow.
Drag, in the context of incompressible potential flow, only arises through induced drag. Without lift, induced drag would also be nil.
To predict lifting flow, vortex sheet modeling is typically used that, by construction, produces non-zero circulation (unless the integral of sheet strength is zero by prescription). An additional prescription of the Kutta condition is required to guarantee a unique physically correct lifting flow.

Image collated from: http://web.mit.edu/16.unified/www/SPRING/fluids/Lecture_Notes/f18.pdf
A mathematically equivalent methodology is to use doublet sheet modeling, using the following tranform:
$$\vec{\gamma}=\hat{n} \times \tilde{\nabla}\mu$$
where $\vec{\gamma}$ is the vortex sheet strength, $\hat{n}$ is the sheet normal, $\mu$ is the doublet strength, and $\tilde{\nabla}$ is the surface-gradient operator.
Somewhat confusingly, a doublet is a combination of a source and sink that are separated by an infinitely small distance. A single doublet is non-lifting. However, a lifting flow can be generated if the doublet sheet varies in strength. This video based on Drela, Flight Vehicle Aerodynamics provides an extremely good intuitive explanation.