TLDR:

for ip in $(pgrep -x evince); do lsof -F +p $ip|grep -i '^n.*.pdf$'|sed s/^n//g; done

Explanation:

Document Viewer is the friendly name for the program /usr/bin/evince. So first we need to find the process ID (PID) of evince:

$ pgrep -x evince

22291

To list all files opened by this PID we will use the lsof command (note that we'll need to repeat this for every PID in case we have more than one instance of evince running)

$ lsof -F +p 22291

some other files opened

.

.

.

n/home/c0rp/File.pdf

Next we'll grep only for pdfs and discard the irrelevant n at start of line:

$ lsof -Fn +p 22291 | grep -i '^n.*.pdf$' | sed s/^n//g

/home/c0rp/File.pdf

Finally to combine everything in one bash line:

for ip in $(pgrep -x evince); do lsof -F +p $ip|grep -i '^n.*.pdf$'|sed s/^n//g; done

This one-liner was inspired from the answer of terdon which is also very interesting in the way it solves the same problem.

If you are interested in what n in lsof -Fn is for, here is quote from man lsof about the -F option:

OUTPUT FOR OTHER PROGRAMS

       When the -F option is specified, lsof produces output that is  suitable

       for  processing by another program - e.g, an awk or Perl script, or a C

       program. 

...

...

       These  are  the  fields  that  lsof will produce.  The single character

       listed first is the field identifier.

...

...

            n    file name, comment, Internet address

...

...

so -Fn, is saying show me file name, comment, Internet address