You'd be deprived of the vitamins and minerals which are found only in vegetables, but you'd live. There are vitamins and minerals in meat and other foods, plenty of them, and your metabolism would adapt to produce the proteins which are missing. Human body is a wonderful organism.
You wouldn't have a higher risk of cancer because cancer is a byproduct of life (the DNA repair mechanism and programmed cell death) and you can't get it by consuming meat or by not consuming vegetables.
You wouldn't gain weight if you're not consuming too much food. The math is simple, if you need 2000 calories, you'll gain weight if you're consuming 2050 calories of vegetables and lose weight if you're consuming 1950 calories of meat.
Your risk of heart disease wouldn't increase.
The only things which would change in this case would be the ones directly related to the nature of carbohydrates, proteins and fat. The carbohydrates in vegetables release energy slowly over a longer period of time, so if you'd eliminate vegetables you'd possibly be hungry more often. However, fat - and the glucose produced from it - would keep you going.
And you might have digestion problems because of lowered intake of fibers until your metabolism adapts.