  Given a sequence $\dot{L}^{\varepsilon}$ of L\'evy noises, we derive
necessary and sufficient conditions in terms of their variances
$\sigma^2(\varepsilon)$ such that the solution to the stochastic heat equation
with noise $\sigma(\varepsilon)^{-1} \dot{L}^\varepsilon$ 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. Rosi\'nski [J. Appl. Probab. 38 (2001) 482-493]
and S. Cohen and J. Rosi\'nski [Bernoulli 13 (2007) 195-210] for
finite-dimensional L\'evy 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
c\`adl\`ag processes with values in a Sobolev space of negative real order.
