As the other answers said already, string players generally avoid playing open strings at all, except for old music in historically informed performance. That's not a hard rule though. If a fast passage can be made easier by playing some of the less emphasized notes on open strings, then I don't think many players would have qualms doing so (those notes just need to be bowed a bit extra carefully not to stick out). In fast passages, especially played by a whole string section, pitch isn't so clearly discernible anyway. And for a passage in something as far out as F♭, the players would probably not have much motivation thinking too much about the mess of flat-signs and fine-intonation details. Most would probably just mentally translate “F♭ = E” at that point, and might, again, use the open string if that makes sense.
Things look differently for notes that are long enough for fine-intonation though. Orchestral players do intonate notes differently depending on context, even if they would be the same on piano. General statements as to whether a flattened note is higher or lower than the sharpened enharmonic are problematic though, because the factors needed to decide are also dependent on the environment. The crucial decision is whether you construct a note from Pythagorean or Ptolemaic tuning. I discussed it somewhat extensively in this answer.
To adress the concrete question
Which is highest in pitch; the middle E♭ in a C minor chord, or the middle D♯ of the penultimate chord of a perfect cadence in E major, played by the same player in a string quartet?
This depends on the interpretation. For both of these notes there are reasons to play them higher than the E♭ on a 12edo piano:
- E♭ in a C-minor chord has the JI ratio of 6:5 to the fundamental, which is about 16ct higher than three 12edo half-steps above C.
- D♯ in the Ⅴ7 of E-major is a leading tone that wants to resolve up. The player may emphasise this effect by intonating the note higher than it would be if the chord were meant as a stable consonance. How much higher isn't really defined, this depends on how much “pain” the player wants to put in the note.
I suspect a first violinist or cellist might play the D♯ a bit higher that the E♭, a 2nd violinist or violist would rather do it the other way around. But more likely, it would just be different for every performance.