Using 172.x.x.x for internal network
It might cause two types of problems:
- You may not be able to access 172.1-15.. or 172.32-255.. IPs, which are possible valid Internet IPs.
- Lot of servers/softwares would come with default configuration for 10.0.0.0/8, 172.16.0.0/12, 192.168.0.0/16 with good trust levels for these internal networks. Those softwares may behave differently for users from say 172.16.. in comparison to users from 172.1... Such problems would be very hard to diagnoze and debug.
If possible shift to 10.0.0.0/8. Only first octet will change rest bytes can remain as it is.
Update:
I read your question wrong, yes, you should fix it to /12, at the very least. I would break the network into smaller networks (subnets), to plan for the future. See Evan's answer to this question to learn about the topic.
You won't be able to access public ip address that are in your /8 subnet (The rest of these are not reserved according to RFC3330) because they will never make it your gateway. The reason is that since the host will think they are on the network, they won't send it to the gateway. If you have a router interface with that mask, then the router will always send requests it receives to that interface because directly connected routes have the lowest administrative distance.
You really do need to fix this, the easiest fix would be just to change the subnet mask on DHCP and any servers to /12 or 255.240.0.0.
Kept for your information, but I read the question wrong the first time:
It won't cause you any problems. 172.16.0.0/12 gives you 1,048,574 ip addresses. However, if you have a lot of hosts, and keep all of them in the same broadcast domain (subnet), you will end up with problems. In all likelihood, if you actually get big enough where that starts to matter you will have a network administrator.
If you wish to learn more, subnetting is the topic you need to read about. Evan wrote a nice introduction to the topic in his answer here.