Do rats eat tomatos off the vine?

I am sure they will eat them. Some critters just bite into them for the juice. I would use a live trap, and then you can relocate them.

Rats are omnivores, meaning they'll eat anything they can get their furry little paws on.

The only way to prevent rodents from chewing on things is a metal barrier. You can find what they call hardware mesh at places like Lowe's/Home Depot and assemble a large cage for your tomatoes. Rats are champion climbers, so you'll want to put a top you can fasten on it.

Get a mesh small enough that they can't get through...perhaps 1/2" in size. You'll want to bury the edge down several inches in the soil to discourage digging.

You probably really don't want these pests anywhere on your property as they chew even on inedible things (like electrical wires or your home's siding) and carry diseases. Set rat traps baited with peanut butter near the foundation of your home and wherever else you think they might be running.

http://homeguides.sfgate.com/keep-rats-off-tomato-plants-50502.html

Traps!

Get some good Rat Poison around. Be warned. Most rat urine carries Leptosporosis or Weils Disease. NASTY

Not an easy solution then

I have heard that rats hate mint.

There are several organic solutions you could try such as:

Peppermint oil as rats rely on their acute sense of smell as their eyesight is so poor and find the minty fragrance of peppermint overpowering. Place a few drops on a piece of cotton wool and situate it around the areas you think the rats may be accessing.

Chilli oil is a common ingredient, known for causing irritation when ingested or inhaled. For this reason, rats and mice tend to steer well clear of areas where chilli oil may be situated.

Rats can’t bear the strong aroma onions let off. Place a cut open onion around the vine and see what effect it has.