János Pintz considered such questions recently, see his preprints [here][1] and [here][2]. In particular, under a weak form of the Elliot-Halberstam conjecture there is an integer $d>0$ such that there are arbitrary long arithmetic progressions of primes $p$ such that $p+d$ is the next prime. Assuming the full conjecture one can take $d\leq 16$, while under a natural strengthening of it one can take any even number $d>0$.

  [1]: http://front.math.ucdavis.edu/1002.2899
  [2]: http://front.math.ucdavis.edu/1004.1067