I am an exchange student in USA right now and I lost my passport. I have to travel back to my country next week and so the Embassy issued me an emergency travel document to travel back. Will exiting USA on an emergency passport cause me problems in travelling back to USA later? I will get a passport made when I get back home but I am really concerned about this issue.
Will exiting USA on an emergency passport cause me problems in travelling back to USA later?
Probably not. There is certainly no legal reason for trouble. There is a possibility that your exit may not be matched to your entry record, but this is an administrative problem, not a legal one. Even so, it would only be a problem if it makes it look like you've overstayed, that is, if you leave before your period of admission expires and return thereafter. This is unlikely for a student, but possible.
If all of these things come to pass, you can correct your records by sending evidence showing you did indeed leave the US to a certain address, so keep your boarding passes and other evidence just in case. You can find more information about that at My sister's passport wasn't stamped when we left the USA - how to rectify?
The main issues are:
ensuring that your I94 record properly records your exit. Check https://i94.cbp.dhs.gov/ a few weeks after you leave
and if you want to return to the USA with the visa that was in your old, lost passport, you’ll need to get that visa re-issued.