Ubuntu – iptables rules stored on Digital Ocean Ubuntu 14.04

I don't know about "Ubuntu", but in Linux generally, "iptables" isn't a service - it's a command to manipulate the netfilter kernel firewall. You can "disable" (or stop) the firewall by setting the default policies on all standard chains to "ACCEPT", and flushing the rules.

iptables -P INPUT ACCEPT
iptables -P OUTPUT ACCEPT
iptables -P FORWARD ACCEPT
iptables -F

(You may need to flush other tables, too, such as "nat", if you've used them)

The following article on the Ubuntu website describes setting up iptables for use with NetworkManager: https://help.ubuntu.com/community/IptablesHowTo

To answer your question succinctly, no: there would not be any "leftover" rules after flushing every table. In the interest of being thorough however, you may want to set the policy for the built-in INPUT and FORWARD chains to ACCEPT, as well:

iptables -P INPUT ACCEPT
iptables -P FORWARD ACCEPT
iptables -P OUTPUT ACCEPT
iptables -t nat -F
iptables -t mangle -F
iptables -F
iptables -X

Clear ip6tables rules:

ip6tables -P INPUT ACCEPT
ip6tables -P FORWARD ACCEPT
ip6tables -P OUTPUT ACCEPT
ip6tables -t nat -F
ip6tables -t mangle -F
ip6tables -F
ip6tables -X

...and that should do it. iptables -nvL should produce this (or very similar) output:

Chain INPUT (policy ACCEPT 0 packets, 0 bytes)
 pkts bytes target     prot opt in     out     source               destination

Chain FORWARD (policy ACCEPT 0 packets, 0 bytes)
 pkts bytes target     prot opt in     out     source               destination

Chain OUTPUT (policy ACCEPT 0 packets, 0 bytes)
 pkts bytes target     prot opt in     out     source               destination