(Harry and Hagrid had entered in a wand shop) For some reason, the back of his neck prickled. The very dust and silence in here seemed to tingle with some secret magic.
(Harry Potter and the Sorcerer's Stone)
Rodet’s 21st Century Thesaurus and Rodet’s Desk Thesaurus say prickle is the synonym of tingle; and the word seems to have to take one who experience the tingling, yet two inanimate objects are preceded. Can this word have the inanimate experiencer?
The very dust and silence in here seemed to tingle with some secret magic.
The word very is a key part of this sentence, which essentially says,
There was so much magic in here, even the dust and the silence seemed to tingle with magic.
Collins lists, among its definitions of very:
used in metaphors to emphasize the applicability of the image to the situation described
In other words, getting back to your intitial question, no, inanimate objects don't usually "tingle" with magic, but this room is so enchanted, that these very ordinary things (dust and silence) seemed to tingle with magic. The fact that the verb tingled is being applied in such an "unusual" way is not a mistake; in fact, it is quite intentional, because it helps the reader picture a room that has an aura of enchantment and wonder.
Also, it's worth noting the significance of the word seemed. When included in a sentence, seem can help stretch the applicability of a verb beyond its normal limits.
The weather was so nice, even the clouds seemed to sing.
The haunted house was so isolated, even its empty rooms seemed to urge us away.
although, when speaking figuratively and metaphorically, words such as very and seemed aren't required, because they can be inferred from the context:
I was so hungry, I could hear the pastries screaming my name.
which doesn't mean I was hallucinating, it simply means:
I was so hungry, it seemed like the very pastries were screaming my name.
I'm not quite sure what you mean by inanimate experiencer but the difference I see with those two words is as such.
Prickle can be used both to describe a sensation and also to describe an object that causes that sensation. E.g. a cactus feels, or is, prickly because it prickles you when you touch it.
Whereas I never hear people say that and object is tingly. You can describe a sensation as tingly but not an object.
Furthermore the idea of something tingling you is an altogether much softer, more gentle sensation than something prickling you. Prickling implies a spike or a sharper sensation whereas tingle is the kind of word you might also use to describe excitement or soft vibrations on the skin.
As we walked into the concert every part of me was tingling with excitement.
This expresses the idea of restless animation flowing through the body.
Prickle can have darker connotations, related to the verb to prick, like when something sharp goes into your arm you can say:
The spine of the rose pricked me
Something that has perhaps a spiky and coarse texture. Think of how English speakers call the Nopal cactus of Mexico the 'Prickly pear'
Prickle is only sometimes a very loose synonym of tingle. The verb usage derives from the noun prickle (a small, sharp pointed object, such as a thorn), and verb to prick (pierce with a sharp point; puncture).
As a verb, prickle has several related meanings and usages, including both intransitive (to stick out or stand up like prickles), and intransitive (to have or feel a pricking or prickling sensation; to tingle, and to affect with a prickling sensation).
It's idle speculation whether in OP's precise context the intended meaning is that the [hairs on the] back of his neck adopted a more "outward-pointing" orientation (as goosebumps - reflex erection of hairs of the skin in response to cold or emotional stress or skin irritation), or that they gave Harry a prickling sensation. Grammatically, either interpretation is possible; semantically, it makes no real difference.
As to whether a "non-conscious" subject, such as the back of the neck, can be used with an "experiential verb" (which would normally imply "awareness" on the part of the subject having the experience), the answer is that parts of the body can, if they have the relevant sensory capability. Thus:
[The snake's] forked tongue tasted the air
My knee hurts
My eyes have seen the glory of the Lord
His body moved, his ears heard, his mouth spoke
etc., etc.
Tingle can be used when talking of a body part.
The cold hair made her face tingle.
The same is true for prickle.
Her eyes prickled with tears.
Tingle with is normally used referred to people, as in "She was tingling with excitement." In the text you quoted, it is used figuratively.
