Receptors split into two broad signalling classes. Ionotropic receptors are themselves ion channels - the agonist binds and the pore conducts ions within milliseconds. Metabotropic receptors couple to G-proteins and act via second messengers over seconds.
Sorting the choices:
$5\text{-HT}_3 \rightarrow$ cation channel (ionotropic)
$\text{Nicotinic} \rightarrow$ cation channel (ionotropic)
$\text{GABA-A} \rightarrow$ chloride channel (ionotropic)
$H_2 \rightarrow G_s\text{-coupled GPCR} \Rightarrow \uparrow cAMP$ (metabotropic)
Since every histamine receptor subtype is G-protein-coupled, $H_2$ is the lone non-channel here.
\[\boxed{H_2 \text{ receptor is NOT a ligand-gated ion channel}}\]