Given a sequence $\dot{L}^{\varepsilon}$ of Lévy noises, we derive necessary and sufficient conditions in terms of their variances $σ^2(\varepsilon)$ such that the solution to the stochastic heat equation with noise $σ(\varepsilon)^{-1} \dot{L}^\varepsilon$ converges in law to the solution to the same equation with Gaussian noise. Our results apply to both equations with additive and multiplicative noise and hence lift the findings of S. Asmussen and J. Rosiński [J. Appl. Probab. 38 (2001) 482-493] and S. Cohen and J. Rosiński [Bernoulli 13 (2007) 195-210] for finite-dimensional Lévy processes to the infinite-dimensional setting without making distributional assumptions on the solutions such as infinite divisibility. One important ingredient of our proof is to characterize the solution to the limit equation by a sequence of martingale problems. To this end, it is crucial to view the solution processes both as random fields and as càdlàg processes with values in a Sobolev space of negative real order.