Assuming you have a 2nd gen 600, the factory rim size is 4.5" and uses a 160/60ZR17 tire. A review of Bridgestones site (OEM tire was a BT56) lists the suggested and acceptable rims sizes as:
160 - 4.5 (4.25 - 5.0)
170 - 5.0 (4.25 - 5.5)
So, the 170 is OK on the 4.5 rim, but the rim is smaller than suggested. This will cause a slightly sharper profile which will pull in a slight bit of the width. At the end of it all, you will end up with a slightly taller and wider rear tire, but not overly obvious.
So...if your doing it for image reasons, I'm not sure you will notice much difference, but it will be a little wider. The difference should not be enough to cause clearance problems with the chain.
To go any wider, you will need a difference rear rim/sprocket arrangement. I assume you can swap in a Bandit 1200 swingarm and rear wheel, but some else here may able to confirm.