When in the key of F-Major would you use C# or Db in notation?

Obviously nothing native to the key, but various borrowed chords and what-not, such as: a V-I tritone substitution of Gb to F would include a Db; augmented 6th chords, such as German or Italian 6ths, built off of Db directly; a borrowed dominant leading to Dm (A7) would include a C#.

EDIT to add: Also altered dominant chords, such as building a diatonic C7 into a C7(b9) would include a Db. Getting further afield, if you borrowed a Bbm into the key of F, you might approach the Bbm with a dominant 7(#5) chord, yielding an F7(#5) with a C#.

Lots of fun possibilities! :-)

The appropriate note depends on context.

For example, if you're using it in the context of a minor plagal cadence, Bbm to F (iv--I) then it's a Db because it's the minor third of Bb minor.

If you're transitioning from F to A7 (i.e. I--III7, common in jazz standards) then it'd be C# because that's the major third of A major.

They may be the same pitch but they are not the same note. A C# in F is a raised fifth, Db is a lowered 6th. Which one you use depends on the context/function of the pitch.

Use C#. Attempting to guess which note to use based on heuristics will just lead to irritated users.