In the beginning of Kizumonogatari Tekketsu-hen,
Araragi is burnt because he is exposed to the sun.
I thought he would become sun tolerant after he is 'cured' of being a vampire since he can walk in the sun without problems in the other seasons.
However, there is a scene in Nekketsu-hen where
he is having a chitchat with Hanekawa out in the open.
Is this some kind of inconsistency? Or did something happen in between?
I'll answer you but I have to say that I will spoil the end of the light novel (and the third animation movie).
In the whole story of Kizumonogatari Araragi is never outside when there is light. If there is light in the movie, it's just a mistake. A reason for Shaft to make something beautiful, that's all.
About the following stories of Monogatari, you will notice that Araragi lost the most part of his vampire abilities. In the end of Kizumonogatari you will learn that Araragi have to kill Shinobu if he wants to turn into a human. That's why Shinobu is so weak in Bakemonogatari. Araragi didn't want to kill Shinobu so they found a compromise. By weakening Shinobu, Araragi became weak as a vampire and lost the most part of his abilities. It also means that his weakenesses, like the sun became less important.
You will notice in bakemonogatari and other stories that Araragi hates the sun, not because he hates it as a human, but because he kept a small part of his weakeness to the sun.
Actually, at the beginning of the Kizumonogatari movie Shinobu looks younger than in Bakemonogatari but that's not the case. This is misleading. In the light novel they say she look like a ten year old girl in Kizumonogatari and a eight year old girl in Bakemonogatari.
This also explain why Araragi is able to regenerate himself so fast in his battle against Kagenui Yozuru. Shinobu was older in this scene. She was a teenager.
I checked on the wiki and here is what they say :
Kiss shot planned on having Koyomi kill her so she could finally find a proper way to die and allow her minion to turn back into a human. Instead, after having her plan revealed, Koyomi rejects it as he doesn't want to kill her. With the help of Meme Oshino, he finds a compromise that leaves no ones wishes granted. Instead of killing her completely, he would get her to the brink of death, weakening her power, making him as close to human as possible. In return, she'd be so weak that she wouldn't even be able to take the same name anymore. To keep her alive, Koyomi would occasionally feed her his blood. This plan goes through. The now unnamed vampire holds a grudge and refuses to speak.
Source : http://bakemonogatari.wikia.com/wiki/Shinobu_Oshino
Watching the scene again, there are two peculiar things that might explain your question.
First is that the whole segment is tinted heavily orange. This might indicate that it either is a dawn or a sunrise. This means there is no direct sunglight. Or it might be artistic way of showing Araragi's perception of night time, as it was stated that Araragi's vampire abilities allow him to see in the dark as if it was daylight.
Second thing to note is that the sky is really cloudy. This again means that there is no direct sunlight.
In either case, the segment is drawn to show clearly that there is no direct sunlight, nor that there is possibility of there being even ray of sunlight. This allows Araragi to walk outside, even if it might seem that it is a day.
