A romantic relationship? Officially, none that we know of. It's possible that any number of them could have boyfriends, girlfriends, or suitors, but we never see it or hear about it in any of the games or official works.
It's worth noting that both Shinki and Suwako have presumably been in a relationship at some point in their lives and may still commit to it, as they have daughters and a descendant respectively. There's also much speculation as to exactly what the relationship between Yuyuko and [Konpaku] Youki's amounted to, in their time as acquaintances though it is most likely thought that Youki [Konpaku] was Yuyuko's previous Gardener.
Un-officially? Well, there's a whole article on Touhouwikia regarding fan shipping. The most popular pairings seem to be (in no particular order):
- Marisa and Alice - Marisa and Patchouli - Marisa and Reimu - Yukari and Reimu - Sakuya and Remilia - Sakuya and Meiling - Keine and Mokou - Kaguya and Mokou - Yuyuko and Youmu
There's tons more; any pairing of characters that a given doujin author would like to see together has probably been done at some point. Those just seem to be the ones that occur most frequently.
It should also maybe be noted that these are all lesbian pairings because Touhou Project has an all-girl cast (Rinnosuke is an exception as of below), not because the characters officially swing that way. (It should also maybe be noted that some of the above characters have been characterized by ZUN with heterosexual interests in official books and such, making a good number of these fan-pairings utter fan-fantasy.)
To be more precise, Touhou Project has an all-girl cast... until you throw Morichika Rinnosuke into the mix. It's also pretty popular among doujin artists and aspiring fans to pair the [many] girls up with Kourindou shopkeeper Rinnosuke, as he is the sole humanoid male character with any canon character development. He is most often paired like so, roughly in order of descending notability:
(most often seen) - Rinnosuke and Keine - Rinnosuke and Marisa - Rinnosuke and Yukari - Rinnosuke and Tokiko
(less often seen) - Rinnosuke and Reimu - Rinnosuke and Sakuya - Rinnosuke and Youmu
Then again, he too is often portrayed as being practically asexual (for somewhat understandable reasons), so remember to take all this with a grain of salt.