Nothing bad will happen if you do use expired Neosporin but the expired Neosporin ointment may not work as good at treating the wound or preventing infection.
The expired Neosporin may also smell bad or be hard and not be effective and might if it's old enough cause a further infection due to potential bacteria.
If your Neosporin ointment is only expired by a few months to a year then it's probably okay to use it on your wound.
However if the Neosporin is expired longer than a year then it would really be better to get a new tube of Neosporin that is fresh.
Expired Neosporin that is old may not get rid of the bacteria and a new tube of Neosporin is not all that expensive especially when your health is concerned.