If you need to read the environment variables of an arbitrary process, the /proc filesystem makes this easy on Linux. The environment variables are shown in /proc/PID/environ:
$ cat /proc/19065/environ DISPLAY=localhost:0.0SHELL=/bin/bashPWD=/home/phil...
In a shell, it will just look like the variables are smooshed together. They're actually separated by \0 (null character), which you can see if you're manipulating this data in some programming language or using a proper text editor:
C-u M-! cat /proc/19065/environ RET
[Update: ps can also be made to show the environment in a manner which is more human-readable but slightly less machine-readable; see comments.]
(Found via a Stack Overflow post about changing another process's environment, which is much more difficult.)