A Green Card application has been filed on my behalf, and the Form I-140 has been approved, can MIT continue to issue Form I-20/Form DS-2019 for me to maintain my F-1 or J-1 student status and authorized F-1/J-1 benefits?

First, we STRONGLY advise that you be sure to have a consultation with an immigration attorney before you take any steps towards applying for a Green Card, and continue to consult with that attorney during the application process.  If you do not have an immigration attorney already, please contact the MIT ISO and we can provide a copy of our immigration attorney referral list for your use.  In many cases, it may not be beneficial to begin the Green Card process while in F-1 or J-1 status, but rather transitioning to H-1B first (which has dual intent).  An immigration attorney can work with you to determine a best strategy.

If you have a Green Card application in process, and you have a Form I-140 or Form I-130 (family-based petition) pending or approved, but have NOT yet filed the Form I-485 for adjustment of status, MIT ISO may still be able to issue the Form I-20/Form DS-2019 and recommend benefits of your F-1 or J-1 status (applying for CPT, OPT, STEM OPT Extension, or J-1 Academic Training; travel signatures; etc.). 

HOWEVER, if you have obtained an employment authorization (EAD) card and/or Advance Parole travel permission based on your Green Card application, and you use either of those items (work using the EAD card, or travel outside the U.S. and re-enter the U.S. using your Advance Parole), then you would no longer be maintaining your F-1 or J-1 status and MIT ISO would need to terminate your SEVIS record for your F-1 or J-1 status.  At that point, your status to remain in the U.S. is dependent upon your pending Green Card application.

This is a complex situation and requires planning.  Your immigration attorney may advise that you NOT travel outside the U.S. while your Green Card application is pending to avoid any risks to your pending application or potential denial of your ability to return to the U.S. (since you have shown by starting the Green Card process as your potential intent to remain permanently in the U.S.).
Again, please consult an immigration attorney before you take any steps towards applying for a Green Card/U.S. Permanent Residency.

If any questions arise, for you or your immigration attorney, please contact your ISO Advisor.