Hobbits are known to be almost permanently barefoot, with their tough feet protecting them from the ground.
However, it seems that not wearing shoes or boots would have its drawbacks. If you walked around enough, especially beyond the well-maintained roads of Hobbiton, you would probably cut your foot or something at least once.
Are there any instances of Hobbits injuring their feet in a non-battle related way? Book canon is preferred, but instances from any official work is fine.
Yup! There was at least one scene in-universe where a hobbit hurt his feet. While it wasn't a cut, it still probably wouldn't have happened if he had been wearing shoes:
The afternoon was turning into evening when he came out again and stumbled and fell in a faint on the 'door-step'. The dwarves revived him, and doctored his scorches as well as they could; but it was a long time before the hair on the back of his head and his heels grew properly again: it had all been singed and frizzled right down to the skin.
This is the scene right after Bilbo Baggins talks to Smaug for the first time in The Hobbit, and Smaug breathes fire at him as he runs away.
Also, in the original trilogy, there were many instances where Frodo and Sam's feet looked pretty bruised and hurt when in Mordor. Not exactly proven, but it is certainly very possible.
While not quite what you had in mind, the closest I can think of is Lalia Clayhanger, wife of the Old Took's grandson Fortinbras II, who was sometimes referred to as "Lalia the Fat." Tolkien writes of her:
She was not at the famous Party (SY 1401), but was prevented from attending rather by her great size and immobility than by her age. [...] Lalia, in her last and fattest years, had the custom of being wheeled to the Great Door, to take the air on a fine morning.
The Letters of J.R.R. Tolkien 214: To A. C. Nunn (draft).
In the real world, obesity has been linked to (among other health effects) a wide array of foot disorders. If Lalia is so overweight that she is unable to move under her own power, it seems likely that she's done some serious damage to her feet and/or ankles.
However, I've not been able to find any reference to foot injuries caused by going barefoot through the countryside. As has been pointed out in comments1, Hobbits are described in the prologue as having tough soles, which would make these sorts of injuries much less likely:
[T]hey seldom wore shoes, since their feet had tough leathery soles
Fellowship of the Ring Prologue 1: "Concerning Hobbits"
SFF.SE's own Darth Melkor once pointed out that Hobbits were capable of wearing shoes (and that Bilbo was intended to have worn boots during the events of The Hobbit), which suggests those sorts of injuries are still possible, though they presumably don't make for very interesting reading.
As revealed in some of the special features on the Lord of the Rings extended edition DVDs, at least two of the film hobbits received foot injuries on-set:
Presumably this is because, unlike their literary counterparts, film-Hobbits' feet only have latex soles, not leather.
1 By NKCampbell and Matt Gutting
Thus it was that Frodo learned how Trotter had tracked Gollum as he wandered southwards, through Fangorn Forest, and past the Dead Marshes, until he had himself been caught and imprisoned by the Dark Lord. 'Ever since I have worn shoes,' said Trotter with a shudder, and though he said no more Frodo knew that he had been tortured and his feet hurt in some way. But he had been rescued by Gandalf and saved from death.
The Return of the Shadow - "In the House of Elrond"
Eventually Tolkien decided to make Trotter a man and cut out the whole arc with the wooden shoes. And then sometime after that Trotter was renamed to Strider.
