Will a hop vine still grow if the tip has been removed?

As Denny said it will reduce your vertical growth. However as the years go on you'll get more shoots coming up than you want to train if you are trying to keep things tidy and growth maximized. At that point you can compete with the birds and eat the hop shoots / tops yourself. If you google you'll find recipes for pickling them and when to harvest. Some of the chefs around here try to get out to the hopyards to pick tips from the extra bines before they are mowed down.

Hopefully someone else can answer about whether you should mow down the current ones this year and hope for more shoots. I'm guessing that is better to do in later years.

Established hops can put out new shoots for a while, and I've read (on freshhops.com, I believe) that commercial growers often cut back the first shoots of the spring and then train the second, hardier set that comes up. So even if your first set gets chewed off, they may yet put up more shoots from the rhizome.

If your hops are first year, you might not get much of a harvest out of them anyway, so don't sweat it too much if they don't grow like crazy. They're just working on getting their roots established.

The bine (not vine) will produce lateral shoots if the tip is broken off, but will not grow any longer.

I too have been babying some hops this year (Glacier/Chinook) and I'm finding the birds are the bane of my existence right now but I have seen some rabbits about. So far, I'm finding that as long as the soil conditions are favorable, the plants bounce back fairly quickly. Once a rhizome has established itself, they'll continue to shoot off new vines. I'm still trying to figure out why the birds are messing with the plants though. It's not as if there's anything there for them to eat. And hops are definitely not native to my location. The nets have kept them at bay for the most part now.