I have installed OS X Mountain Lion as a VM in VMware Fusion (version 4.1.2) for environment testing. The host OS is OS X 10.7.4, and is connected to the internet via a wired connection (in fact I get the same problem if it is connected via wireless as well).
As long as VMware Fusion is running, my internet for the Host OS stops working, but remains functional inside the VM.
I am using NAT for VMware Fusion.
For additional testing, I have an Ubuntu box on Virtual Box. When VMware Fusion is off, I have internet on both my host OS and the Ubuntu VM, and when VMware fusion is running, I have no internet connection except inside VMware Fusion.
VMware Fusion blocks internet even if the actual VM is suspended.