Local anesthetics sting at first for quite some people, but then after a few minutes there should be no feeling, neither of pain from the injections or from the procedure. The sting is likely due to the fact that in addition to blocking the molecules that allow nerves to transmit impulses, they activate irritant receptors (the ones that respond to hot pepper, garlic, etc.). It's kind of a race to see which happens first--whether the irritant receptors are activated before the nerves can't fire anymore and you wouldn't know anyway.
I suspect that in some of us on the spectrum, these irritant receptors may be prone to being hypersensitive. Maybe we are more prone to feeling this initial burning, then. There's also the fact that some anesthetic injections also have epinephrine (adrenaline) in them to constrict blood vessels. I remember once as a kid I had a minor oral surgery and felt jittery for maybe 5-10 minutes after the injection, more than I had earlier due just to the anxiety about the procedure. Maybe for some people the vessel constriction spreads far enough out of the numbed area that it can be felt as pain.
Once the numbness sets in, though, it should be nearly complete. I remember when I had tongue surgery, the injections into the tongue felt kind of like it was getting "zapped", but afterward even cutting in that super-sensitive area was completely imperceptible, aside from the pressure of the tools on the other parts of the mouth.