You can use opensnoop from BCC, which uses eBPF under the hood:
# ./opensnoop -p 1576
PID COMM FD ERR PATH
1576 snmpd 11 0 /proc/sys/net/ipv6/conf/lo/forwarding
1576 snmpd 11 0 /proc/sys/net/ipv6/neigh/lo/base_reachable_time_ms
1576 snmpd 9 0 /proc/diskstats
1576 snmpd 9 0 /proc/stat
1576 snmpd 9 0 /proc/vmstat
[...]
This is quite performant since it uses kprobes instead of having to restart syscalls, like strace does.
You can also do this with strace (potentially with -f to follow the traced process' children), but its way of operating, involving restarting syscalls as part of ptrace will slow down your application somewhat:
# strace -e open -p 15735
open("/usr/lib/locale/locale-archive", O_RDONLY|O_CLOEXEC) = 3
open("/usr/lib/gconv/gconv-modules.cache", O_RDONLY) = -1 ENOENT (No such file or directory)
open("/usr/lib/gconv/gconv-modules", O_RDONLY|O_CLOEXEC) = 3
open("/usr/lib/python2.7/site-packages", O_RDONLY|O_NONBLOCK|O_DIRECTORY|O_CLOEXEC) = 4
open("/usr/lib/locale/locale-archive", O_RDONLY|O_CLOEXEC) = 3
open("/etc/localtime", O_RDONLY|O_CLOEXEC) = 8
[...]
You can also start your application this way if desired, using strace [executable], or strace -f [executable].