If you have SystemTap, you can monitor the entire host for files being opened.

[root@localhost tmp]# cat mon
#!/usr/bin/env stap
probe syscall.open { printf ("pid %d program %s opened %s\n", pid(), execname(), filename) }
# ./mon
pid 14813 program touch opened "/etc/ld.so.cache"
pid 14813 program touch opened "/lib64/libc.so.6"
pid 14813 program touch opened 0x7f7a8c6ec8d0
pid 14813 program touch opened "foo2"
[...]
#

Per Chris Down, you can use strace -p to examine an already-running process, to see what files it opens from now onwards.

If you want to see files opened for the entire duration of a process, right from the start, use strace with the executable name. Adding -f ensures that any forked sub-processes also get reported. Example

# strace -e open -f /bin/id
open("/etc/ld.so.cache", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libselinux.so.1", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libc.so.6", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libpcre.so.1", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libdl.so.2", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libpthread.so.0", O_RDONLY|O_CLOEXEC) = 3
open("/usr/lib/locale/locale-archive", O_RDONLY|O_CLOEXEC) = 3
open("/proc/thread-self/attr/current", O_RDONLY|O_CLOEXEC) = -1 ENOENT (No such file or directory)
open("/proc/self/task/1581/attr/current", O_RDONLY|O_CLOEXEC) = 3
open("/usr/share/locale/locale.alias", O_RDONLY|O_CLOEXEC) = 3
open("/usr/share/locale/en_US.UTF-8/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/usr/share/locale/en_US.utf8/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/usr/share/locale/en_US/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/usr/share/locale/en.UTF-8/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/usr/share/locale/en.utf8/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/usr/share/locale/en/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/etc/nsswitch.conf", O_RDONLY|O_CLOEXEC) = 3
open("/etc/ld.so.cache", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libnss_files.so.2", O_RDONLY|O_CLOEXEC) = 3
open("/etc/passwd", O_RDONLY|O_CLOEXEC) = 3
open("/etc/group", O_RDONLY|O_CLOEXEC)  = 3
open("/etc/group", O_RDONLY|O_CLOEXEC)  = 3
uid=0(root) gid=0(root) groups=0(root) context=unconfined_u:unconfined_r:unconfined_t:s0-s0:c0.c1023
+++ exited with 0 +++
#

Per Chris Down, you can use strace -p to examine an already running process, to see what files it opens from now until the time you terminate strace or the process itself finishes.

If you want to see files opened for the entire duration of a process, right from the start, use strace with the executable name. Adding -f ensures that any forked sub-processes also get reported. Example

# strace -e open -f /bin/id
open("/etc/ld.so.cache", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libselinux.so.1", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libc.so.6", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libpcre.so.1", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libdl.so.2", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libpthread.so.0", O_RDONLY|O_CLOEXEC) = 3
open("/usr/lib/locale/locale-archive", O_RDONLY|O_CLOEXEC) = 3
open("/proc/thread-self/attr/current", O_RDONLY|O_CLOEXEC) = -1 ENOENT (No such file or directory)
open("/proc/self/task/1581/attr/current", O_RDONLY|O_CLOEXEC) = 3
open("/usr/share/locale/locale.alias", O_RDONLY|O_CLOEXEC) = 3
open("/usr/share/locale/en_US.UTF-8/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/usr/share/locale/en_US.utf8/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/usr/share/locale/en_US/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/usr/share/locale/en.UTF-8/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/usr/share/locale/en.utf8/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/usr/share/locale/en/LC_MESSAGES/coreutils.mo", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/etc/nsswitch.conf", O_RDONLY|O_CLOEXEC) = 3
open("/etc/ld.so.cache", O_RDONLY|O_CLOEXEC) = 3
open("/lib64/libnss_files.so.2", O_RDONLY|O_CLOEXEC) = 3
open("/etc/passwd", O_RDONLY|O_CLOEXEC) = 3
open("/etc/group", O_RDONLY|O_CLOEXEC)  = 3
open("/etc/group", O_RDONLY|O_CLOEXEC)  = 3
uid=0(root) gid=0(root) groups=0(root) context=unconfined_u:unconfined_r:unconfined_t:s0-s0:c0.c1023
+++ exited with 0 +++
#

Using lsof to see what files a process currently has open

# lsof -p $(pidof NetworkManager)
COMMAND   PID USER   FD      TYPE             DEVICE  SIZE/OFF     NODE NAME
NetworkMa 722 root  cwd       DIR              253,0       224       64 /
NetworkMa 722 root  rtd       DIR              253,0       224       64 /
NetworkMa 722 root  txt       REG              253,0   2618520   288243 /usr/sbin/NetworkManager
NetworkMa 722 root  mem       REG              253,0     27776    34560 /usr/lib64/libnss_dns-2.17.so
[...]
#