It depends on specific device and its architecture.
I'm not sure if there are any non-Android-based mainstream eInk readers - most of the ones I heard of (which is not a lot) seem to be Android under the covers and thus can be opened up (e.g. see Kobe Glo).
There is a project, called PRS+, for older Sony PRS devices (PRS-300, PRS-505, PRS-600, PRS-350, PRS-650, PRS-950).
KOReader is a document viewer for Linux based E Ink devices. Currently there are versions for Kindle, Kobo, PocketBook, and Android devices.
The installation procedure depends on the device. Support forums are available on Mobileread.
XDA Developers don't have the e-Ink Kindle, but they have a page for the Nook Touch.
Check out the "Android Development" thread.
There is a crowdfunding campaign for an e-ink device that says they will release the firmware as free software.
We intend on making the firmware for the 13.3 inch e-reader available to the entire community.
I have no affiliation with the campaign or the people behind it.