What is the minimal symmetry required for a spin Hamiltonian to describe a spin-liquid ground state?
The definition of a spin liquid as a spin system "with no spontaneously broken symmetries" is out of date and no longer used, partially for the reason you describe. If you perturb as spin-liquid Hamiltonian by adding small terms that break all the symmetries, then the ground state will still be a spin liquid even though there are no longer any symmetries that could possibly be broken. Moreover, a spin liquid actually can spontaneously break symmetries; see the third paragraph of http://arxiv.org/abs/1112.2241.
The more modern definition of a spin liquid is a spin system with "intrinsic topological order." This can be defined in many equivalent ways (at least for a gapped system - the gapless case raises more subtle issues): (a) the inability to be deformed into a product state by local unitary operations, (b) nonzero topological entanglement entropy, (c) low-energy physics that can be described by a topological quantum field theory, (d) excitations with anyonic statistics, etc.
tparker's answer is absolutely correct, but it's worth noting why the "old fashioned" definition was still useful. According to the higher dimensional extension of the Lieb-Schultz-Mattis theorem by Hastings and others, a gapped system with (a) translational invariance (b) an odd number of S=1/2 moments per magnetic unit cell, and (c) unbroken SO(3) symmetry must be a spin-liquid in the sense he described (anyons). So the old definition was a sufficient, but not necessary, condition for spin-liquid physics.
One way to rephrase your question is: how much symmetry is required for a Lieb-Schultz-Mattis theorem to still hold? For example, Oshikawa and Hastings showed that you can break down $SO(3) \to U(1)$ (rotation invariance about one axis), and the theorem still holds at zero-magnetization. Later work showed you can breakdown $SO(3) \to \mathbb{Z}_2 \times \mathbb{Z}_2$, or you even break $SO(3)$ completely if you keep time-reversal invariance. These two are probably the minimal cases in the sense you're asking.