Check if IP Forwarding is enabled

Using sysctl:

sysctl net.ipv4.ip_forward
net.ipv4.ip_forward = 0

Enable IP Forwarding on the fly

As with any sysctl kernel parameters we can change the value of net.ipv4.ip_forward on the fly (without rebooting the system):

sysctl -w net.ipv4.ip_forward=1

Permanent setting using /etc/sysctl.conf

If we want to make this configuration permanent the best way to do it is using the file /etc/sysctl.conf where we can add a line containing net.ipv4.ip_forward = 1

/etc/sysctl.conf:
net.ipv4.ip_forward = 1

if you already have an entry net.ipv4.ip_forward with the value 0 you can change that 1.

To enable the changes made in sysctl.conf you will need to run the command:

sysctl -p /etc/sysctl.conf

On RedHat based systems this is also enabled when restarting the network service:

service network restart

and on Debian/Ubuntu systems this can be also done restarting the procps service:

/etc/init.d/procps.sh restart

Using distribution specific init scripts

Although the methods presented above should work just fine and you would not need any other method of doing this, I just wanted to note that there are also other methods to enable IP Forwarding specific to some Linux distributions.
For example Debian based distributions might use the setting:

/etc/network/options:
ip_forward=no

set it to yes and restart the network service.
Also RedHat distributions might set this using:

/etc/sysconfig/network:
FORWARD_IPV4=true

and again restart the network service.

Regardless the method you have used once you have completed this you can check it out using the same method shown above:

sysctl net.ipv4.ip_forward
net.ipv4.ip_forward = 1

cat /proc/sys/net/ipv4/ip_forward
1

If the result is 1 then the Linux system will start forwarding IP packets even if they are not destined to any of its own network interfaces.