Ketosis is starvation mode.
Mmm, not exactly, cos there's no such thing as starvation mode. And you could easily be in ketosis and be extremely well nourished, for example if you were doing a low carb diet.
While you've got plenty of fat to burn your body isn't going to behave as though it's starving - cos it isn't, it can burn excess fat. It's only when you haven't got as much fat to burn that you're at risk of losing lean muscle mass in the absence of enough calories. That's why you need to move up to 810 when you reach a BMI of 25.0.
But Becks is right in that in the early days of ketosis your body might not be able to convert fat into energy fast enough - it takes at least three weeks to adapt to ketosis. That's one of the reasons why strenuous exercise isn't recommended in the early days of this diet. But people can and do run marathons in a state of ketosis (think you might need more than 450 calories worth of low carb food if you were planning that, though.

).
VLCDs are more about calorie deficit than ketosis anyway. On any diet there will be a reduction in metabolic rate but on a VLCD it doesn't really apply because the calorie intake is so low - you can't help but lose weight.
So, to cut to the chase, exercise won't take you out of ketosis. But there is a risk it might make you hungry, so be careful not to overdo it and leave yourself vulnerable to a raid on the fridge.
Finally, if you're hoping exercise will speed up your losses, be aware it doesn't seem to work that way on a VLCD. Exercise for fitness, not increased weight loss.