Is it correct to say "I fancy your photos"? If yes, what would that phrase mean? How different would the meaning be from "I like your photos"? In what context would "I fancy your photos" sound natural?
"I fancy your photos" would be appropriate to say if you were saying it to a photographer, who will take it as a compliment. It would sound natural in that context only.
In any other context, it would be best to say "I like your photos" or "You look nice in your photos"
If you fancy something, it means you desire it, you want to have it.
Either this is used in terms of physical attraction, if you fancy a person, it means you desire them and find them attractive. If you fancy doing something, it means you desire to do it. If you fancy an object, then it means you would like to acquire it.
If you say "I fancy your photos", that could either mean you find the people in them attractive; or you wish to keep them. Either way it would be very strange phrasing. Best to stick with "I like your photos".
"Fancy," used as a verb CAN mean to desire. It can also be used as a synonym for "like."
A quick trip to the dictionary reveals this definition: "to have a fancy for : like."
And the noun's definition, just to check the verb's: "a liking formed by caprice rather than reason : inclination "
(Note: There are other definitions for both the noun and the verb.)
So, in the sentence "I fancy your photos" I would say that it is equivalent to "I like your photos (for no reason that I can think of.)"
Generally speaking, the word fancy isn't used in this sense much any more, at least in the US. It's a little old-fashioned, and some would say rustic. That said, I think the only place it would sound natural used like this would be in a work of fiction.
Also beware that @Orbling's answer details a completely legitimate meaning of the word fancy, which can make the meaning of a sentence like "I fancy your photos" ambiguous, especially without context.