The standard concordance model of the Universe is based on the cosmological constant as the driver of accelerating expansion. This concordance model is being subjected to a growing range of interlocking observations. In addition to using generic observational tests, one can also design tests that target the specific properties of the cosmological constant. These null tests do not rely on parametrizations of observables, but focus on quantities that are constant only if dark energy is a cosmological constant. We use supernova data in null tests that are based on the luminosity distance. In order to extract derivatives of the distance in a model-independent way, we use Gaussian processes. We find that the concordance model is compatible with the Union 2.1 data, but the error bars are fairly large. Simulated data sets are generated for the Dark Energy Survey supernova survey and we show that this survey will allow for a sharper null test of the cosmological constant if we assume the Universe is flat. Allowing for spatial curvature degrades the power of the null test.