As far as I know, 10minutemail does not do this. I personally use 33mail for this sort of forwarding.
However these are designed around obscuring MY email address, and using 10minutemail.com or 33mail.com's domains.
I need to create emails for my customers that will forward to THEIR email addresses when another party contacts them, preferably using my own domain.
While I guess I technically could figure out a way to use something like 10minutemail and register accounts for all of the users of my site, it would probably require:
1. the users of my site to tell 10minutemail that it's ok to forward to them (during 10minutemail's registration process)
2. i would have to post 10minutemail.com domains as the contact info for the customers of my site, which is an undesirable UI.
If I'm missing something and I can in fact use existing services to do what I want, please fill me in, because I want this, I just don't want to build it (yet, anyways).