How are presynaptic burst firing signals transmitted post-synaptically?
Short answer
It depends.

Background
It all depends on the neuronal circuit and neurotransmitter system under investigation. For example, raphe neurons release 5-HT in a burst-like fashion onto cortical dendrites. However, the postsynaptic effect is inhibitory, so no action potentials there (Gartside, 2000).

Intracortical recordings in the rat have shown that each action potential in a burst resulted in neurotransmitter (glutamate) release and subsequent excitatory postsynaptic currents (EPSCs) and potentials (EPSPs) (Williams &amp; Stuart, 1999). So that would positively answer your question, at least in this specific neuronal circuit.

Reference
- Gartside, Neurosci (2000); 98(2): 295-300
- Williams &amp; Stuart, J Physiol (1999); 521(2): 467–82