IDK if I would. Probably yes if it's not problematic, contradictory to the themes of the story, or makes my fan base toxic because of it. (The last thing I would want would be a fan base that would attack each other and even me over the smallest thing like me not telling the story I don't want to tell or me making a harmless joke that they took umbrage with.)
Though I have always been wary about Shippers given how it seems that most toxicity comes fro them. (Looking at you Voltron and Adventure Time!) Like, I always worry they might ardently ship two characters together that I think have absolutely no right or even should be together. (Like, I remember when I was first working on DRGN/DRAGOONS, I was super worried people would ship the two protagonists together if only because they had a rivalry with each other when I would actually be horrified at such a development given that the two basically honest-to-goodness wanted to murder each other at the start of the story due to how different their ideologies were. So you could see how a ship between the two "because they are angst rivals and that totally secretly love each other!" could be quite problematic in general if that makes sense to you you guys.)
Other than that, I'd embrace most fan fiction and fan art whole-heartedly. Though part of me (often jokingly) worries about the types of fan art that may spawn from Alpha Alpaca and DRAGOONS given that both star casts of anime-esque colorful neon animal-like humanoid characters. Glances in the direction of Sonic the Hedgehog, Digimon, and Pokemon and shudders in horror.
(Though said series could also spawn some really cute/good fan art, so that I would look forward to...)