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While having a dental surgery, I've got this question that why I can still talk, open/close my mouth, move my lips and so on while I can't feel anything at all in my mouth?
If the local anesthesia interrupts the alert that goes through brain, how is the muscle still functional? Is it some sort of one-way interruption which only disables the alert that go "from" the nerve "to" the brain? Or are there different nerves for this task?
@Nicolai's answer is not entirely correct.
Background
The most common local anesthetics are all the "-caine" drugs - like novocaine, lidocaine, - and even cocaine has some of the same pharmacology, as well as other effects (in movies this is why cops might rub some white powder on their lips - if it's cocaine, it will make their lips tingle; this is not standard training for non-movie police; also not advised for drug sellers or purchasers, because for exactly this reason cocaine is often cut with other -caines or other compounds that give a tingly feeling).
What all of these drugs do is to block voltage-gated sodium channels. These are the channels that propagate action potentials through axons and excitable dendrites. Other local anesthetics that aren't used in the clinic, like tetrodotoxin from pufferfish, block the same types of channels. At sufficient concentrations, local anesthetics block all nervous activity: unlike @Nicolai's answer, they do shut down nerves (as a brief note for terminology: nerves are bundles of axons containing many many individual fibers).
So back to your original question: how come the muscle isn't impaired during dental surgery?
Geometry of innervation of the face
Take a look at this picture of the nerves innervating the facial muscles and the muscles themselves: https://bmc.med.utoronto.ca/cranialnerves/wp-content/images/c_07/Facial-vii12_labelled768.jpg (note: I wasn't able to find a good image that has a license allowing me to reproduce it here; if someone finds one and wants to edit it in, please do - thanks!)
They mostly come in from the sides of the face, and so they are further away from the site of local anesthetic injection. The muscles themselves also extend further away. It may be that the muscle is partly anesthetized, but you don't realize it.
Limited diffusion and location of injections
Diffusion of local anesthetics is limited because they actually act inside cells, and only the unionized form can diffuse across cell membranes. Once inside the cell, the tend to be ionized and therefore get "stuck". Therefore, they don't typically diffuse nearly as far as other compounds of similar size, especially lipophilic compounds.
The nerve endings that sense pain, on the other hand, have to be at the surface of the skin/mucous membranes. That is, they are exactly where the local anesthetic is being injected. The objective of local anesthesia is often to inject enough to affect only those surface tissues, trying to spare the deeper tissues.
Effects on nerves
However, it isn't too uncommon to overdo it and to paralyze the underlying muscles or nerves. Even nerves far away can be affected, for example see this paper discussing cases where the local anesthetic travels near the eyes and impairs eye movement. There are also other non-dental procedures where block of a particular nerve is the intended result: this can be seen as an alternative to general anesthesia. One example is the use of an epidural (essentially into the space around the spinal cord) injection during childbirth or surgery. The extent of paralysis depends on the dose, however, and usually the intent is to avoid major paralysis and use other drugs like opiates to assist in preventing pain.
The answer to this question boils down to 'because local anesthesia does not interrupt the nerves themselves'.
From the wiki article on local anesthetics (LA):
All LAs are membrane-stabilizing drugs; they reversibly decrease the rate of depolarization and repolarization of excitable membranes (like nociceptors). Though many other drugs also have membrane-stabilizing properties, not all are used as LAs (propranolol, for example). LA drugs act mainly by inhibiting sodium influx through sodium-specific ion channels in the neuronal cell membrane, in particular the so-called voltage-gated sodium channels. When the influx of sodium is interrupted, an action potential cannot arise and signal conduction is inhibited.
This means that the LAs don't actually shut down nerves. What they shut down is the ability of nerves to sense mechanical movement. Because mechanical movement on the cellular level is a bit different from what we usually associate with it, this means that in an area affected by LAs the nerves can not report feelings of pain, touch or temperature [actually also spicy taste] to the brain. All other nerve functions (including muscle control) are not affected
@Bryan's answer isn't totally correct.
Short Answer: There are different nerves. Motor neurons and sensory neurons are not identical and anesthesia affects them differently.
When dentist inject novacaine they are targeting the V3 or V2 nerve trunks, where both sensory and motor neuron bundles are in close proximity. Take a look at this illustration for clarification https://operativeneurosurgery.com/lib/exe/fetch.php?w=400&tok=5831b0&media=trigeminal_nerve.jpg The injection site is usually near the jaw at the side of the face, this allows the whole side of either zone be sufficiently numb. So it must be a difference in the physical makeup of the neurons.
Some of the differences between motor and sensory neurons are: Motor neurons are long, multipolar, and contain efferent fiber, sensory neurons are short, unipolar, and contain afferent fibers. Myelin is a protective sheath of proteins and lipids that protect and insulate the nerve fibers. Myelination is the process in which myelin is formed over the nerve fibers. But afferent and efferent fibers undergo different types of myelination. Progression of anesthesia is related to the diameter of affected never fibers, conduction velocity, and myelination. The difference in myelination between the two types of fibers is the main reason anesthetics mostly affect sensory neurons.
